Question
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?
Answers
Answer
should bee 4/9 i think
Answered by
GPT-4o mini
To find the ratio of the areas of the two triangles described, we first need to calculate the area of each triangle.
### Area of the Smaller Triangle:
The smaller triangle is formed by the points:
- \( (5, 9) \)
- \( (5, 3) \)
- \( (2, 3) \)
**Base**: The distance between \( (5, 3) \) and \( (2, 3) \) is:
\[
5 - 2 = 3
\]
**Height**: The distance from \( (5, 3) \) to \( (5, 9) \) is:
\[
9 - 3 = 6
\]
**Area**: The area of a triangle is given by the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substituting the values:
\[
\text{Area}_{\text{small}} = \frac{1}{2} \times 3 \times 6 = \frac{1}{2} \times 18 = 9
\]
### Area of the Larger Triangle:
The larger triangle is formed by the points:
- \( (12, 6) \)
- \( (12, 2) \)
- \( (10, 2) \)
**Base**: The distance between \( (12, 2) \) and \( (10, 2) \) is:
\[
12 - 10 = 2
\]
**Height**: The distance from \( (12, 2) \) to \( (12, 6) \) is:
\[
6 - 2 = 4
\]
**Area**: Using the formula for the area of a triangle again:
\[
\text{Area}_{\text{large}} = \frac{1}{2} \times 2 \times 4 = \frac{1}{2} \times 8 = 4
\]
### Ratio of the Areas:
Now we can find the ratio of the area of the smaller triangle to the area of the larger triangle:
\[
\text{Ratio} = \frac{\text{Area}_{\text{small}}}{\text{Area}_{\text{large}}} = \frac{9}{4}
\]
Thus, the ratio of the area of the smaller triangle in relation to the area of the larger triangle is:
\[
\frac{9}{4}
\]
### Area of the Smaller Triangle:
The smaller triangle is formed by the points:
- \( (5, 9) \)
- \( (5, 3) \)
- \( (2, 3) \)
**Base**: The distance between \( (5, 3) \) and \( (2, 3) \) is:
\[
5 - 2 = 3
\]
**Height**: The distance from \( (5, 3) \) to \( (5, 9) \) is:
\[
9 - 3 = 6
\]
**Area**: The area of a triangle is given by the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substituting the values:
\[
\text{Area}_{\text{small}} = \frac{1}{2} \times 3 \times 6 = \frac{1}{2} \times 18 = 9
\]
### Area of the Larger Triangle:
The larger triangle is formed by the points:
- \( (12, 6) \)
- \( (12, 2) \)
- \( (10, 2) \)
**Base**: The distance between \( (12, 2) \) and \( (10, 2) \) is:
\[
12 - 10 = 2
\]
**Height**: The distance from \( (12, 2) \) to \( (12, 6) \) is:
\[
6 - 2 = 4
\]
**Area**: Using the formula for the area of a triangle again:
\[
\text{Area}_{\text{large}} = \frac{1}{2} \times 2 \times 4 = \frac{1}{2} \times 8 = 4
\]
### Ratio of the Areas:
Now we can find the ratio of the area of the smaller triangle to the area of the larger triangle:
\[
\text{Ratio} = \frac{\text{Area}_{\text{small}}}{\text{Area}_{\text{large}}} = \frac{9}{4}
\]
Thus, the ratio of the area of the smaller triangle in relation to the area of the larger triangle is:
\[
\frac{9}{4}
\]
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