Question

The area of a triangle is found by multiplying half of the measure of its base by its height. The area of the following right triangle is 18 square centimeters.

A. Write and solve an equation to find the value of h. Include all of your work in your answer.

B. In two or more complete sentences, verify your answer using mental math.

50 points

Answer 1

Below!

Step-by-step explanation (a):

Given:

• Area of Triangle = 18 cm²
• Base of Triangle = 6 cm
• Height of (Triangle = (h + 1) cm

Formula:

`\text{Area of triangle} = (1)/(2) * \text{Base} * \text{Height}`

Substitute the measures into the formula:

`:\implies 18 \ \text{cm}^(2) = (1)/(2) * 6 * \text{(h} + 1\text{)}`

SImplify the R.H.S:

`:\implies18 \ \text{cm}^(2) = 3 * \text{(h} + 1\text{)}`

Divide both sides by 3:

`:\implies (18)/(3) = \frac{3 * \text{(h} + 1\text{)}}{3}`

Simplify both sides:

`:\implies 6 = \text{(h} + 1\text{)}`

Subtract 1 both sides:

`:\implies 6 - 1 = \text{h} + 1- 1`

Simplify both sides:

`:\implies 5= \text{h}`

Step-by-step explanation (b):

Reviewing on how to verify our answer:

We can verify our answer by substituting the value of "h" in (h + 1) to determine the height of the triangle. Then we can substitute the height and the base in the "area of triangle" formula (1/2 x Base x Height).

• Note: Keep in mind that the evaluated area should be 18 cm², as said in the question. If the area evaluated is different from 18 cm², then the value of "h" is incorrect.

Verifying our answer through mental math:

• Clearly, the height of the triangle is 6 cm, as when "h" is substituted in the expression (h + 1), we get 5 + 1.
• As said above, the formula for determining the area of a triangle is 1/2 x Base x Height. Clearly, half of 6 is 3.
• Lastly, when 6 (Base) and 3 (Half of the height) are multiplied, we get 18 cm², which is the area stated in the question.

Thus, the value of "h" is verified through mental math.

Answer 2

`\sf Area\:of\:a\:triangle=\frac12 * base * height`

Given:

• area = 18 cm²
• base = 6 cm
• height = (h + 1) cm

Substitute the given values into the equation and solve for h:

`\sf \implies 18 = \sf \frac12 * 6 * (h+1)`

`\sf \implies 18 = \sf \frac62 (h+1)`

`\sf \implies 18 =3(h+1)`

expand using distributive property of addition
`a(b+c)=ab+ac`:

`\sf \implies 18=3h+3`

subtract 3 from both sides:

`\sf \implies 3h=15`

divide both sides by 3:

`\sf \implies h=5`

To verify:

Half of the base is 3 cm.

If h=5, then the height of the triangle is 6 cm.

Multiplying 3 by 6 is 18.

This matches the given area, so we can verify that h = 5.