ABCD is a square of side 8 cm. PQR is a triangle with side QR equal to the diagonal ,of the square. ST is the segment joining mid-points of sides PQ and PR of triangle PQR. An isosceles triangle XYZ is drawn right-angled at Y such that the ‘length of the perpendicular YO drawn from Y on hypotenuse XZ equals to ST. What is the area of triangle XYZ? (in sq cm)

Options

A. 128
B. 96
C. 64
D. 32

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript and break it down into clear steps.

### Step 1: Calculate the diagonal of square ABCD
The side of the square ABCD is given as 8 cm. The formula for the diagonal (d) of a square with side length (s) is:
\[ d = s \sqrt{2} \]
Substituting the value of s:
\[ d = 8 \sqrt{2} \text{ cm} \]

### Step 2: Determine the length of side QR of triangle PQR
Since QR is equal to the diagonal of the square ABCD, we have:
\[ QR = 8 \sqrt{2} \text{ cm} \]

### Step 3: Find the length of segment ST
ST is the segment joining the midpoints of sides PQ and PR of triangle PQR. The length of ST is half of QR:
\[ ST = \frac{QR}{2} = \frac{8 \sqrt{2}}{2} = 4 \sqrt{2} \text{ cm} \]

### Step 4: Understand the isosceles triangle XYZ
Triangle XYZ is isosceles and right-angled at Y. The length of the perpendicular YO from Y to the hypotenuse XZ is equal to ST:
\[ YO = ST = 4 \sqrt{2} \text{ cm} \]

### Step 5: Relate the sides of triangle XYZ
In an isosceles right triangle, if the lengths of the two equal sides are denoted as A, then the relationship between the legs and the hypotenuse (XZ) can be expressed as:
\[ XZ = A \sqrt{2} \]
The area of triangle XYZ can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base and height are both equal to A.

### Step 6: Calculate the area of triangle XYZ
To find A, we use the relationship involving the height (YO):
In a right-angled isosceles triangle, the height (YO) can also be expressed as:
\[ YO = \frac{A}{\sqrt{2}} \]
Setting this equal to the known value of YO:
\[ \frac{A}{\sqrt{2}} = 4 \sqrt{2} \]

### Step 7: Solve for A
Multiplying both sides by \(\sqrt{2}\):
\[ A = 4 \sqrt{2} \times \sqrt{2} = 4 \times 2 = 8 \text{ cm} \]

### Step 8: Calculate the area using the value of A
Now substituting A back into the area formula:
\[ \text{Area} = \frac{1}{2} \times A \times A = \frac{1}{2} \times 8 \times 8 = \frac{64}{2} = 32 \text{ cm}^2 \]

### Final Answer
The area of triangle XYZ is:
\[ \text{Area} = 32 \text{ cm}^2 \]

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