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Math  /  Geometry

QuestionInstructions Solve the following problem and select your answer from the choices given.
Question
The area of the triangle above is 21 . What is the value of xx ? 3 6 7 11

Studdy Solution

STEP 1

1. The triangle is a standard triangle, not necessarily a right triangle.
2. The area of the triangle is given as 21 square units.
3. The base of the triangle is x+1 x + 1 .
4. The height of the triangle is x x .
5. We need to find the value of x x .

STEP 2

1. Write the formula for the area of a triangle.
2. Substitute the given values into the area formula.
3. Solve the resulting equation for x x .
4. Verify the solution by checking if it satisfies the given area.

STEP 3

The formula for the area of a triangle is given by:
Area=12×base×height \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

STEP 4

Substitute the given values into the area formula. The base is x+1 x + 1 and the height is x x , and the area is 21:
21=12×(x+1)×x 21 = \frac{1}{2} \times (x + 1) \times x

STEP 5

Solve the equation for x x :
21=12×(x+1)×x 21 = \frac{1}{2} \times (x + 1) \times x
Multiply both sides by 2 to eliminate the fraction:
42=(x+1)×x 42 = (x + 1) \times x
Expand the right side:
42=x2+x 42 = x^2 + x
Rearrange the equation to form a standard quadratic equation:
x2+x−42=0 x^2 + x - 42 = 0

STEP 6

Solve the quadratic equation x2+x−42=0 x^2 + x - 42 = 0 using factorization:
Look for two numbers that multiply to −42-42 and add to 11. The numbers are 77 and −6-6.
Factor the quadratic equation:
(x+7)(x−6)=0 (x + 7)(x - 6) = 0
Set each factor equal to zero and solve for x x :
x+7=0orx−6=0 x + 7 = 0 \quad \text{or} \quad x - 6 = 0
x=−7orx=6 x = -7 \quad \text{or} \quad x = 6
Since x x represents a length, it must be positive, so x=6 x = 6 .

STEP 7

Verify the solution by substituting x=6 x = 6 back into the area formula:
Base = x+1=6+1=7 x + 1 = 6 + 1 = 7
Height = x=6 x = 6
Calculate the area:
Area=12×7×6=12×42=21 \text{Area} = \frac{1}{2} \times 7 \times 6 = \frac{1}{2} \times 42 = 21
The calculated area matches the given area, confirming that the solution is correct.
The value of x x is:
6 \boxed{6}

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