Honors Geometry A Unit 7: Triangle Relationships
What are the coordinates of the intersection points of the medians with the sides of the triangle?
(4, 2), (4, −4), and (−2, 2)
An isosceles triangle has two sides of length nn where nn is an integer greater than 11. What is the smallest possible integer length of the third side?
1
You are given that one angle of a triangle is 100°. Can you find the longest side of the triangle?
A triangle's angles sum to 180°, so the other angles of the triangle must sum to 80°. It follows that the other angles cannot be greater than 100°, so 100° is the largest angle. Thus, the side opposite the given angle is the longest side.
Aaron draws the triangle below and states that the length of DE is 6 inches. Is he correct? Why or why not?
Aaron is incorrect. DE does not bisect AB and BC, so the Triangle Midsegment Theorem cannot be used to determine the length of DE.
If the perimeter of a triangle is 22 cm and each side length is a whole number, what is the greatest length that any one of the sides can be?
According to the Triangle Inequality Theorem, the sum of any two sides of a triangle must be larger than the third side. So, the greatest length that any one of the sides can be is 10 cm. The other two sides would sum to 12 cm, which is larger than 10 cm. But, if one of the sides were 11 cm, then the other two sides would sum to 11 cm, which is not greater than the third side length.
What does the Inequality In One Triangle Theorem say about equilateral triangles?
Equilateral triangles have three angles that are all congruent, so the sides of the triangle are all congruent.
John is having trouble finding the medians of a triangle in the coordinate grid. Explain to him the best method.
Find the midpoint of each side using the formula (x1+x22,y1+y22)x1+x22,y1+y22. Then, draw segments from each vertex to the midpoint on the opposite side.
State the converse of the Triangle Inequality Theorem.
If one side of a triangle is at least as long as the sum of the other two sides, the triangle does not exist.
What is the converse of the hinge theorem?
If two sides of one triangle are congruent to two sides of another triangle, and the third side of one triangle is longer than the third side of the other triangle, then the angle between the congruent sides in the triangle with the larger third side is larger than the angle between the congruent sides in the other triangle
Is it possible for a triangle with a pair of congruent angles to not be an isosceles triangle?
It is not possible. A triangle with a pair of congruent sides always has congruent base angles, and vice versa. So, a triangle with a pair of base angles will always be an isosceles triangle or an equilateral triangle (which is a special type of isosceles triangle.)
Justine says she can prove that not all isosceles triangles have congruent base angles by showing that a right triangle with two legs that are congruent does not have congruent base angles. Is Justine correct?
Justine is not correct. The base angles of an isosceles right triangle are the angles where the hypotenuse intersects the two legs. Because the legs are congruent, these angles are also congruent.
For △PQR△PQR with QR>PQQR>PQ, for the first step toward proving ∡QPR>∡PRQ∡QPR>∡PRQ using the inequality in one triangle theorem, point SS is placed on side QR¯¯¯¯¯¯¯¯QR¯, and line segment PS¯¯¯¯¯¯¯PS¯ is drawn. Explain the two line segments that must be congruent.
Line segments PQ¯¯¯¯¯¯¯¯PQ¯ and QS¯¯¯¯¯¯¯QS¯ must be congruent so that ∠QPS∠QPS is congruent to ∠QSP∠QSP.
Ryan says that the Triangle Mid-Segment Theorem states that the segment joining the midpoints of two sides of a triangle is parallel to and half the length of the third side. So, it follows that if a segment is half the length of the third side of a triangle, it is a mid-segment of the triangle. Is Ryan correct?
No, he is not correct. It is true only if the segment is parallel to the third side.
Does it matter how wide you set the compass when constructing the first two arcs?
The width of the first set of arcs does not matter as long as it is greater than half the length of BC. The second set of arcs must be created with the same compass width as the first set.
Triangle Mid-Segment
Theorem states that the segment joining the midpoints of two sides of a triangle is parallel to and half the length of the third side.
Is the given statement true or false? Explain. The medians of a triangle lie inside the triangle.
The statement is true. The medians of a triangle will always be inside the triangle.
Which angles in the following triangle are the base angles?
Angles A and C.
Suppose an engineer is placing a brace across the midsegment of a bridge truss that is 12 feet wide. How long must the brace be cut?
By the Triangle Midsegment Theorem, the segment joining the midpoints of the two sides is half the length of the third side. Half of 12 is 6, so, the brace must be 6 feet long.
An architect decides that all the doors in a house must be the same width, and the distance between the latch of an open door and the connecting doorway must be 6 feet. If this distance is obtained by a door open at a 100°100° angle, is it possible to create this large of an opening at a 90°90° angle? Explain.
No, it is not possible to create an opening of 6 feet if the angle is 90°90°. By the Hinge Theorem, the distance of the opening at 90°90° must be smaller than 6 feet.
Can you use the Triangle Inequality Theorem to find the length of unknown sides of a triangle? Explain.
No, the Triangle Inequality Theorem can give only a range of side lengths.
Does the Triangle Mid-Segment Theorem apply only to right triangles? Explain.
No, the Triangle Mid-Segment Theorem is proved using a right triangle, but it applies to all triangles.
A contractor is designing a triangular deck with side lengths of 10 feet, 5 feet, and 5 feet. Assess and evaluate the information to determine if the design is possible? Explain why or why not.
No, the design is not possible. Since two of the side lengths add up to the length of the third side, the measurements do not form a triangle by the Triangle Inequality Theorem.
Do you need all three medians to locate the centroid?
No, you need to have only two medians. Because they all intersect at a single point, you can find the centroid by finding the intersection point of any two medians.
A triangle has two angles that are equal in measure and a third that is not equal to the other two. Can you find the longest side? Use reasoning to explain your answer.
No. If the two congruent angles are greater than 60°, then the sides opposite them are equal and they will be the longest sides. If they are less than 60°, then the side opposite the third angle is the longest side.
For △PQR△PQR with PR>PQPR>PQ, for the first step toward proving ∡PQR>∡PRQ∡PQR>∡PRQ using the inequality in one triangle theorem, on which side should point SS be placed?
PR
Determine the missing part of the proof of the statement |PR|+|QR|>|PQ||PR|+|QR|>|PQ| for triangle △PQR△PQR. Begin by drawing a triangle PRSPRS such that △PRS△PRS is an isosceles triangle and line segment SQSQ passes through RR. Per the base angles theorem, ∠S≅∠SPR∠S≅∠SPR. Since ∠SPQ>∠SPR∠SPQ>∠SPR, it must be true that ∠SPQ>∠S∠SPQ>∠S. [______________] Notice that SQSQ can be written as the sum of segments RSRS and QRQR. So |RS|+|QR|>|PQ||RS|+|QR|>|PQ|. Since △PRS△PRS is an isosceles triangle, RS=PRRS=PR. Thus, by substitution, |PR|+|QR|>|PQ||PR|+|QR|>|PQ|.
Since the side of a triangle opposite a larger angle is larger in length, it must be true that |SQ|>|PQ||SQ|>|PQ|.
What is the first step to finding the centroid of a triangle?
Sketch circles at the vertices that are at least half the length of the adjacent sides. Use these to find the medians.
Consider a triangle ABC where m∠A=65°m and m∠B=30°. Order the sides from least to greatest.
The angles of a triangle must sum to 180°, so angle C measures 85°. The side opposite angle B is the smallest side, the side opposite angle A is the second largest side, and the side opposite angle C is the largest side. Thus, the sides listed in order from least to greatest is AC, BC, AB.
Jonas says that the triangle below proves that the base angles of an isosceles triangle are not always congruent because angles A and C are not congruent. What is wrong with Jonas's argument?
The base angles of isosceles triangles are those opposite the congruent sides. So, in this case, the base angles are actually angles B and C.
When proving |LM|+|MN|>|LN||LM|+|MN|>|LN| in △LMN△LMN, one possible first step is to extend MN¯¯¯¯¯¯¯¯¯¯MN¯ to a point PP. Explain how far MN¯¯¯¯¯¯¯¯¯¯MN¯ should be extended.
The line segment needs to be extended to PP so that LM=MPLM=MP because the result is ∠MLP≅∠MPL∠MLP≅∠MPL.
centroid
The point of concurrency of the medians of a triangle
What does the Triangle Midsegment Theorem state?
The segment joining the midpoints of two sides of a triangle is parallel to and half the length of the third side.
The longest side of a triangle is 15 cm. If the lengths of the other two sides are 2x+32x+3 and 3x+23x+2, what must each side be greater than in order to form a triangle?
The side labeled 2x+32x+3 must be greater than 7 cm, and the side labeled 3x+23x+2 must be greater than 8 cm.
What is the length of DE in the figure below? Note that triangle ABC is a right triangle.
Use the Pythagorean Theorem to find the side length of AC. a2+b2AB2+AC262+AC236+AC2AC2AC======c2BC2102100648a2+b2=c2AB2+AC2=BC262+AC2=10236+AC2=100AC2=64AC=8 Now, use the length of side AC to find the length of side DE. Since, DE bisects sides AB and BC, it follows that side DE and AC are parallel and that side DE is half the length of side AC. DE===12AC1284DE=12AC=128=4 Thus, side DE has a length of 4 units.
Carlos uses both sides of a ruler to draw two parallel lines and then draws a triangle, with one of the lines as the base and the other intersecting two sides of the triangle. He notices that the base of the triangle is not twice the length of the other segment. Where do you think Carlos's error is?
When Carlos drew the triangle, he did not draw it such that one of the parallel lines is a mid segment of the triangle. He either drew the triangle too tall or too short.
Does the figure below provide enough information to conclude that DE is a mid-segment?
Yes, because DE is a bisector of AC and parallel to AB, it also bisects BC.
Nina says that she needs to find only two medians in order to find the centroid. Do you agree?
Yes, because all three medians intersect at a single point, only two medians are needed to find the intersection point.
Does the triangle inequality theorem apply to △WXY△WXY? Explain.
Yes, because the triangle inequality theorem applies to all triangles.
Is it possible to have an isosceles triangle that is also a right triangle? If so, what do you know about that triangle?
Yes, it is possible. The congruent angles must each measure 45°45°.
If the side of a triangle opposite a 75°75° angle is 7 cm, is it possible for the side of a triangle opposite a 60°60° angle to also be 7 cm?
Yes, since it is not stated that the sides adjacent to the known angles are congruent, the Hinge Theorem does not apply. So, it is possible for the indicated sides to be the same length.
Is it possible to draw a segment within a triangle that is half the length of the base, but not the mid segment?
Yes, you can draw a segment that is not parallel to the base, but is still half the length. That segment is thus not a mid segment.
isosceles triangle
a triangle with at least two equal sides.
Inequality In One Triangle Theorem
if two sides of a triangle are unequal, then the angle opposite the longer side is larger than the angle opposite the shorter side.
base angles
isosceles triangle are the angles formed by the base and one leg of the triangle. They are opposite the two sides that are congruent. For triangle ABC below angles B and C are the base angles.
To determine the point at which medians of a triangle intersect one another, it is necessary to determine the equations of the lines for two medians. The coordinates of which two points are used in the calculation of the equation for a given median?
one vertex of a triangle and the midpoint of its opposite side
Triangle Inequality Theorem
states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Hinge Theorem
that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of one triangle is larger than the included angle of the other triangle, then the third side of the triangle with the larger angle is longer than the third side of the other triangle. AB≅DE, AC≅DF, and ∠CAB>∠FDE∠CAB>∠FDE. We want to prove that BC>EF
Triangle Inequality Theorem states
that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This theorem, along with its converse, can be used to solve problems.
Base angles of an isosceles triangle
the angles formed by the base and one leg of the triangle
characteristic of an isosceles triangle is that
the base angles of the triangles are congruent.
median
the middle score in a distribution; half the scores are above it and half are below it
Do the three segments of lengths 3 ft, 5 ft, and 6 ft form a triangle? Explain.
yes Use the Triangle Inequality Theorem to determine if the three side lengths form a triangle. 3+58>>66✓3+5>68>6✓ 3+69>>56✓3+6>59>6✓ 5+611>>33✓5+6>311>3✓ All three inequalities are satisfied, so a triangle exists with side lengths of 3 ft, 5ft, and 6ft.
Suppose there are two triangles, △JKL△JKL and △STU△STU, where JL=STJL=ST, KL=TUKL=TU, and JK<SUJK<SU. What does the converse of the hinge theorem say about these two triangles?
∡L<∡T
