In the standard (x, y) coordinate plane, a triangle has vertices at A(-2, -3), B(6, 1), an… — ACT

In the standard $(x, y)$ coordinate plane, a triangle has vertices at $A(-2, -3)$ , $B(6, 1)$ , and $C(1, 6)$ . The length of the altitude from vertex $C$ to side $AB$ can be written in the simplified radical form $a\sqrt{b}$ , where $a$ and $b$ are positive integers. What is the value of $a + b$ ?

Cevap: 8

Cevap

The final answer is 8.

The altitude length of the triangle is

3\sqrt{5}

. In this simplified radical form, the coefficient

a

is 3 and the radicand

b

is 5. Summing these values gives

3 + 5 = 8

Adım Adım Çözüm

Calculate the area of triangle

ABC

using the shoelace formula or by bounding the triangle in a rectangle.

Area = 30

The area is needed to determine the altitude length using the area formula of a triangle.

Calculate the length of the base side

AB

using the distance formula.

AB = 4\sqrt{5}

The base length is required to solve for the height perpendicular to it.

Set up the triangle area formula

Area = \frac{1}{2} \times \text{base} \times \text{height}

to find the height

h

h = 3\sqrt{5}

This solves for the length of the altitude from vertex

C

to side

AB

Identify

a

and

b

from the simplified radical form

a\sqrt{b} = 3\sqrt{5}

and calculate

a + b

To provide the final requested sum.

Anahtar Kavram

Calculating the altitude of a triangle in the coordinate plane by relating its area and side lengths.

Soru

Cevap

Adım Adım Çözüm

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