Two sides of a triangle are 6 cm and 8 cm. If height of the triangle corresponding to 6 cm side is 4 cm, find :
(i) area of the triangle
(ii) height of the triangle corresponding to 8 cm side.

Answer

To solve the problem step by step, we will first find the area of the triangle and then calculate the height corresponding to the 8 cm side.

### Step 1: Calculate the Area of the Triangle

The formula for the area of a triangle is given by:

\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \]

In this case, we can take the base as the side measuring 6 cm and the height corresponding to this base is given as 4 cm.

Substituting the values into the formula:

\[ \text{Area} = \frac{1}{2} \times 6 \, \text{cm} \times 4 \, \text{cm} \]

Calculating the area:

\[ \text{Area} = \frac{1}{2} \times 24 \, \text{cm}^2 = 12 \, \text{cm}^2 \]

### Step 2: Calculate the Height of the Triangle Corresponding to the 8 cm Side

To find the height corresponding to the 8 cm side, we will use the area we just calculated. We know the area of the triangle is also given by:

\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \]

Now we will take the base as 8 cm and let the height corresponding to this side be \( h \).

Using the area we found:

\[ 12 \, \text{cm}^2 = \frac{1}{2} \times 8 \, \text{cm} \times h \]

Now, we can solve for \( h \):

\[ 12 = 4h \]

Dividing both sides by 4:

\[ h = \frac{12}{4} = 3 \, \text{cm} \]

### Final Answers:

(i) The area of the triangle is \( 12 \, \text{cm}^2 \).

(ii) The height of the triangle corresponding to the 8 cm side is \( 3 \, \text{cm} \).

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