Rectangle PQRS has a perimeter equal to 34 units. What is x, the length of the diagonal PR?
Hint
Is ∠PQR a right angle? What's the length of PQ?
Answer
x = 12.2 units
Example 2
What is the length of the diagonal x?
Hint
Are the two quadrilaterals congruent? Are you sure?
Answer
x = 5 units
Example 3
Prove that if the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
Hint
Show ΔPQS ≅ ΔQPR. What can we can say about angles that are both congruent and supplementary?
Answer
Statements
Reasons
1. PQRS is a parallelogram
Given
2. PR ≅ QS
Given
3. PS ≅ RQ
Opposite sides of parallelograms are congruent (1)
4. PQ ≅ PQ
Reflexive property
5. ΔPQS ≅ ΔQPR
SSS Postulate (2, 3, 4)
6. ∠SPQ and ∠PQR are supplementary
Consecutive angles of parallelograms are supplementary (1)
7. ∠SPQ and ∠PQR are congruent
CPCTC (5)
8. m∠SPQ + m∠PQR = 180°
Definition of supplementary angles (6)
9. m∠SPQ = m∠PQR
Definition of congruent angles (7)
10. 2(m∠SPQ) = 180°
Substitution (8, 9)
11. m∠SPQ = 90°
Division property of equality (10)
12. ∠SPQ is a right angle
Definition of a right angle (11)
13. PQRS is a rectangle
A parallelogram with at least one right angle is a rectangle (1, 12)
Example 4
WXYZ is a rectangle. Prove that ΔWQZ is isosceles.
Hint
Show WQ ≅ ZQ.
Answer
An isosceles triangle is one with two congruent sides. So we want to show two sides are congruent, and WQ and ZQ look like good suspects. If we start using the fact that the diagonals of a rectangle are equal, we know WY ≅ XZ. Because WXYZ is a rectangle, it is also a parallelogram, which means that its diagonals bisect each other. In other words, WQ ≅ YQ and ZQ ≅ XQ. Since WY ≅ XZ and they are bisected into equal parts, we know that WQ must be congruent to ZQ. Since these two legs of ΔWQZ are congruent, the triangle is isosceles. Whew!
Example 5
The plan for a truss calls for cross beams BD and CF. How long are these beams together?
Hint
You're looking at a rectangle. Maybe you should be looking at two triangles.
Answer
10 feet
Example 6
If both diagonals of a rectangle total 34 meters and one of the sides is 8 meters long, what is the perimeter of the rectangle?
Hint
You may want to take a second to recognize your oldest and most loyal frenemy: Pythagoras.
