Question
Maura is creating a quilt using right triangles. A large right triangle with a side length of √40/12
centimeters has smaller right triangles with side lengths of 5√3 centimeters along it. How many smaller lengths are running along the longer length?(1 point)
centimeters has smaller right triangles with side lengths of 5√3 centimeters along it. How many smaller lengths are running along the longer length?(1 point)
Answers
GPT-4o mini
To determine how many smaller right triangles with side lengths of \( 5\sqrt{3} \) centimeters align along the longer side of the large right triangle, we first need to calculate the length of the longer side of the large right triangle.
The side length of the large right triangle is given as \( \frac{\sqrt{40}}{12} \) centimeters.
1. **Simplify \( \frac{\sqrt{40}}{12} \)**:
\[
\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}
\]
Thus,
\[
\frac{\sqrt{40}}{12} = \frac{2\sqrt{10}}{12} = \frac{\sqrt{10}}{6} \text{ centimeters}
\]
2. **Determine how many smaller triangles fit along this length**:
The side length of each smaller triangle is \( 5\sqrt{3} \) centimeters.
To find how many smaller triangles fit, we calculate:
\[
\text{Number of smaller triangles} = \frac{\text{Length of the longer side}}{\text{Length of each smaller triangle}} = \frac{\frac{\sqrt{10}}{6}}{5\sqrt{3}}
\]
This simplifies to:
\[
= \frac{\sqrt{10}}{6 \cdot 5\sqrt{3}} = \frac{\sqrt{10}}{30\sqrt{3}} = \frac{\sqrt{10}}{30\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{30}}{90}
\]
Since \( \frac{\sqrt{30}}{90} \) is a non-integer, it indicates that no whole smaller triangles can fit along this length.
Thus, the answer is that **0 smaller triangles** fit along the longer side of the large triangle.
The side length of the large right triangle is given as \( \frac{\sqrt{40}}{12} \) centimeters.
1. **Simplify \( \frac{\sqrt{40}}{12} \)**:
\[
\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}
\]
Thus,
\[
\frac{\sqrt{40}}{12} = \frac{2\sqrt{10}}{12} = \frac{\sqrt{10}}{6} \text{ centimeters}
\]
2. **Determine how many smaller triangles fit along this length**:
The side length of each smaller triangle is \( 5\sqrt{3} \) centimeters.
To find how many smaller triangles fit, we calculate:
\[
\text{Number of smaller triangles} = \frac{\text{Length of the longer side}}{\text{Length of each smaller triangle}} = \frac{\frac{\sqrt{10}}{6}}{5\sqrt{3}}
\]
This simplifies to:
\[
= \frac{\sqrt{10}}{6 \cdot 5\sqrt{3}} = \frac{\sqrt{10}}{30\sqrt{3}} = \frac{\sqrt{10}}{30\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{30}}{90}
\]
Since \( \frac{\sqrt{30}}{90} \) is a non-integer, it indicates that no whole smaller triangles can fit along this length.
Thus, the answer is that **0 smaller triangles** fit along the longer side of the large triangle.