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- The figure above shows the dimensions of an isosceles triangle in terms of x. What is the area of the triangle?

- 24
- 30
- 48
- 60
- 96

Answer: D

2x+3 = 3x-2 [sides opposite to equal angles]

x = 5

Equal sides = 13

Height: 12

Using Pythagoras theorem we get, base of each half-triangle = 5

Using the property that the perpendicular to the base of an isosceles triangle bisects the base: , we get the base = 10

Area = ½ x Base x Height = ½ x 10 x 12 = 60

- If the perimeter of an isosceles triangle is 40, then which of the following CANNOT be the length of the unequal side?

- 21
- 19
- 17
- 15
- 13

Answer: A

Let the 2 equal sides be x each and the unequal side be y
Perimeter: 2x + y = 40
Now, in any triangle the sum of two sides must be greater than the third side
So x+y >y and x+x > y
Or the sum of the equal sides is greater than the unequal side.
Which means the sum of the equal sides is greater than 40/2 = 20
In other words, y < 20

In the figure above, If ABCD is a square and AE = EB, then which of the following statements is NOT true?

Select __all__ that apply.

- DE = CE
- p = t
- q = s
- r = s
- r = 30

Answer: D, E

Using Pythagoras theorem:

In ∆AED AE^{2} + AD^{2} = DE^{2}

In ∆BEC BE^{2} + BC^{2} = CE^{2}

But AD = BC [sides of a square] and AE = BE [Given]

So DE = CE A is TRUE

DE = CE so the angles opposite must also be equal p = t B is TRUE

q=90-p and s = 90 – t so q = s C is TRUE

r = s if BE = BC but BE = ½ BC. D is FALSE

We know in a 30-60-90 triangle, if r = 30 then BE = √3BC but BE = ½ BC. E is FALSE.

- If √x = 3 and √y = -2, then what is the value of x
^{2}+ y^{2}+2xy in terms of y?

- 25
- 49
- 97
- 121
- 169

Answer: E

√x = 3 so x = 9

√y = -2 so y = 4

x^{2} + y^{2} +2xy = (x+y)^{2} = 13^{2} = 169

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