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Answer:
- KY = 12
- PK = 12
- m∠YKZ = 90°
- m∠PZR = 67°
- AQ = 3
- m∠APQ = 45°
- m∠MNP = 90°
- PM = 6
Explanation:
The diagonals of a rhombus are perpendicular bisectors of each other.
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1. KY is the long leg of right triangle KYR, whose hypotenuse is given as 13 and short leg as 5. If you do not recognize this as a 5-12-13 right triangle, you can find the longer leg using the Pythagorean theorem.
KY² +RK² = RY²
KY² = RY² -RK² = 169 -25 = 144
KY = √144 = 12
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2. K is the midpoint of diagonal PY, so PK = KY = 12.
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3. As we have said, the diagonals cross at right angles.
m∠YKZ = 90°
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4. PZ is parallel to RY, so the alternate interior angles created by diagonal ZR are congruent.
m∠PZR = m∠YRZ = 67°
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5. We recognize that hypotenuse PQ of right triangle PQA is √2 times the side length AP. So, this is a "special right triangle" that is isosceles. The sides and hypotenuse have the ratios 1 : 1 : √2.
AQ = AP = 3
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6. As in any isosceles right triangle, the acute angles are 45°.
m∠APQ = 45°
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7. Each of the smaller acute angles is 45°. The diagonals bisect the corner angles, so each corner angle is 2×45° = 90°.
m∠MNP = 90°
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8. A is the midpoint of PM, so PM is twice the length of AP.
PM = 2×AP = 6