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An illustration shows the first quadrant of a coordinate plane. The x-axis and the y-axis range from 0 to 15. Two right triangles are graphed on the plane. One triangle is formed by connecting points plotted at coordinates left parenthesis 5 comma 9 right parenthesis, left parenthesis 5 comma 3 right parenthesis, and left parenthesis 2 comma 3 right parenthesis. The sides of this triangle are formed by solid lines. The second triangle is formed by connecting points plotted at coordinates left parenthesis 12 comma 6 right parenthesis, left parenthesis 12 comma 2 right parenthesis, and left parenthesis 10 comma 2 right parenthesis. The sides of this triangle are formed by dotted lines.

What is the ratio of the area of the smaller triangle in relation to the area of the larger triangle?

1 answer

To find the ratio of the areas of the two triangles described, we first need to calculate the area of each triangle.

  1. Calculating the area of the smaller triangle: The vertices of the smaller triangle are at (5, 9), (5, 3), and (2, 3).

    • The base of this triangle is the line segment between (5, 3) and (2, 3), which has a length of: \[ \text{Base} = 5 - 2 = 3 \]
    • The height of this triangle is the vertical distance from (5, 3) to (5, 9), which is: \[ \text{Height} = 9 - 3 = 6 \]
    • The area \( A \) of a triangle is given by the formula: \[ A = \frac{1}{2} \times \text{Base} \times \text{Height} \] Substituting in the values we calculated: \[ A_{\text{small}} = \frac{1}{2} \times 3 \times 6 = \frac{18}{2} = 9 \]

  2. Calculating the area of the larger triangle: The vertices of the larger triangle are at (12, 6), (12, 2), and (10, 2).

    • The base of this triangle is the line segment between (12, 2) and (10, 2), which has a length of: \[ \text{Base} = 12 - 10 = 2 \]
    • The height of this triangle is the vertical distance from (10, 2) to (12, 6), which is: \[ \text{Height} = 6 - 2 = 4 \]
    • Again, using the area formula: \[ A_{\text{large}} = \frac{1}{2} \times 2 \times 4 = \frac{8}{2} = 4 \]

  3. Finding the ratio of the areas: The ratio of the area of the smaller triangle to the area of the larger triangle is given by: \[ \text{Ratio} = \frac{A_{\text{small}}}{A_{\text{large}}} = \frac{9}{4} \]

Thus, the ratio of the area of the smaller triangle in relation to the area of the larger triangle is \(\frac{9}{4}\).