Question
Aylissa plans to use four triangles like the one shown to form a diamond shape.
The legs lengths are 3 cm, and 4 cm , the slope is 6 cm
For her plan to work, the triangle must be a right triangle. Determine whether the triangle is a right triangle. Show your work.
The legs lengths are 3 cm, and 4 cm , the slope is 6 cm
For her plan to work, the triangle must be a right triangle. Determine whether the triangle is a right triangle. Show your work.
Answers
GPT-4o mini
To determine if the triangle with leg lengths 3 cm and 4 cm and a slope (hypotenuse) of 6 cm is a right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that for a right triangle with legs of length \(a\) and \(b\), and hypotenuse \(c\), the following relationship holds:
\[
a^2 + b^2 = c^2
\]
In this case:
- \(a = 3\) cm (one leg)
- \(b = 4\) cm (the other leg)
- \(c = 6\) cm (the hypotenuse)
Now we can calculate \(a^2 + b^2\) and \(c^2\):
\[
a^2 = 3^2 = 9
\]
\[
b^2 = 4^2 = 16
\]
Now, add these values together:
\[
a^2 + b^2 = 9 + 16 = 25
\]
Next, we calculate \(c^2\):
\[
c^2 = 6^2 = 36
\]
Now, we compare \(a^2 + b^2\) with \(c^2\):
\[
a^2 + b^2 = 25
\]
\[
c^2 = 36
\]
Since \(25 \neq 36\), the condition of the Pythagorean theorem is not satisfied, and therefore, the triangle with legs 3 cm and 4 cm and a hypotenuse of 6 cm is **not** a right triangle.
\[
a^2 + b^2 = c^2
\]
In this case:
- \(a = 3\) cm (one leg)
- \(b = 4\) cm (the other leg)
- \(c = 6\) cm (the hypotenuse)
Now we can calculate \(a^2 + b^2\) and \(c^2\):
\[
a^2 = 3^2 = 9
\]
\[
b^2 = 4^2 = 16
\]
Now, add these values together:
\[
a^2 + b^2 = 9 + 16 = 25
\]
Next, we calculate \(c^2\):
\[
c^2 = 6^2 = 36
\]
Now, we compare \(a^2 + b^2\) with \(c^2\):
\[
a^2 + b^2 = 25
\]
\[
c^2 = 36
\]
Since \(25 \neq 36\), the condition of the Pythagorean theorem is not satisfied, and therefore, the triangle with legs 3 cm and 4 cm and a hypotenuse of 6 cm is **not** a right triangle.