Prove that a parallelogram is a square iff its diagonals are both congruent and perpendicular
1 answer:
Consider the parallelogram shown below.
The lengths of the sides are a and b.
The lengths of the diagonals are 2x and 2y.
Because the diagonals are both congruent and perpendicular, therefore there are two right triangles as shown.
Note that x = y.
Because x = y, each right triangle is isosceles and has the angles 90°, 45° and 45°.
From the Pythagorean theorem,
For one right triangle,
a² = x² + y² = x² + x² = 2x².
For the other right triangle,
b² = y² + x² = x² + x² = 2x²
Therefore
a² = b²
a = b
It follows that all sides of the parallelogram are equal and each angle is
45+45 = 90°
Therefore the parallelogram is a square.
The lengths of the sides are a and b.
The lengths of the diagonals are 2x and 2y.
Because the diagonals are both congruent and perpendicular, therefore there are two right triangles as shown.
Note that x = y.
Because x = y, each right triangle is isosceles and has the angles 90°, 45° and 45°.
From the Pythagorean theorem,
For one right triangle,
a² = x² + y² = x² + x² = 2x².
For the other right triangle,
b² = y² + x² = x² + x² = 2x²
Therefore
a² = b²
a = b
It follows that all sides of the parallelogram are equal and each angle is
45+45 = 90°
Therefore the parallelogram is a square.
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Answer:
see explanation
Step-by-step explanation:
Given the 3 equations
3x + 5y + 5z = 1 → (1)
x - 2y = 5 → (2)
2x + 4y = 11 → (3)
Use (2) and (3) to solve for x and y
Multiply (2) by 2
2x - 4y = 10 → (4)
Add (3) and (4) term by term
4x = 21 ( divide both sides by 4 )
x =
Substitute this value of x into (3)
2 × + 4y = 11
+ 4y = 11 ( subtract
from both sides )
4y = ( divide both sides by 4 )
y =
Substitute the values of x and y into (1) and solve for z
3 × + 5 ×
+ 5z = 1
+
+ 5z = 1
+ 5z = 1 ( subtract
from both sides )
5z = - ( divide both sides by 5 )
z = -
Solution is
x = , y =
, z = -