Question

A right angle triangle ABC (

• A3.76 cm
• B3.01 cm
• C4.12 cm
• D5.24 cm

Answer 1

Answer:

A. 3.76 cm

To find the length of $AD$, we can use the properties of right-angled triangles and similar triangles when an altitude is drawn to the hypotenuse.

Given Data:

• $\angle B = 90^\circ$

• Side $AB = 8\text{ cm}$

• Side $BC = 15\text{ cm}$

• $BD \perp AC$

Step-by-Step Calculation:

1. Find the length of the hypotenuse ($AC$): Using the Pythagorean theorem in $\triangle ABC$:

   
   

   $AC^2 = AB^2 + BC^2$

   $AC^2 = 8^2 + 15^2$

   $AC^2 = 64 + 225 = 289$

   $AC = \sqrt{289} = 17\text{ cm}$

2. Use the similarity property (or projection rule): When a perpendicular $BD$ is dropped from the right-angled vertex to the hypotenuse, the smaller triangle $\triangle ADB$ is similar to the main triangle $\triangle ABC$ ($\triangle ADB \sim \triangle ABC$).

   
   

   From this similarity, the ratio of the sides gives the relation:

   
   

   $\frac{AD}{AB} = \frac{AB}{AC}$

3. Cross-multiplying gives:

   $AB^2 = AD \times AC$

4. Substitute the known values to solve for $AD$:

   
   $8^2 = AD \times 17$

   $64 = 17 \times AD$

   $AD = \frac{64}{17}\text{ cm}$

Answer:

The length of $AD$ is $\frac{64}{17}\text{ cm}$ (or approximately $3.76\text{ cm}$).