Question
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Answers
Answer
Maria constructed the angle bisector of angle ABC by constructing ray BD.
Maria then constructed the angle bisector of DBC by constructing ray BE.
If the measurement of angle ABC is 68 degrees, what is the measurement of angle EBC?
Maria then constructed the angle bisector of DBC by constructing ray BE.
If the measurement of angle ABC is 68 degrees, what is the measurement of angle EBC?
Answer
Which set of side lengths do NOT form a triangle?
2,3,5
3,5,7
5,7,10
6,14,18
2,3,5
3,5,7
5,7,10
6,14,18
Answer
Carrie constructed AB using a compass and straightedge.
She wants to construct a copy of AB using point C as one of her endpoints. What step does she need to take next?
She wants to construct a copy of AB using point C as one of her endpoints. What step does she need to take next?
Answered by
GPT-4o mini
Yes, that's correct! This is known as the Pythagorean theorem. In a right triangle, if we denote the lengths of the two legs (the sides that form the right angle) as \(a\) and \(b\), and the length of the hypotenuse (the side opposite the right angle) as \(c\), the theorem states:
\[
c^2 = a^2 + b^2
\]
This relationship is fundamental in geometry and is used to find the lengths of the sides of right triangles, as well as in various applications across mathematics, physics, engineering, and more.
\[
c^2 = a^2 + b^2
\]
This relationship is fundamental in geometry and is used to find the lengths of the sides of right triangles, as well as in various applications across mathematics, physics, engineering, and more.
Answered by
GPT-4o mini
To determine the measurement of angle \( EBC \), we can use the properties of angle bisectors.
1. **Step 1:** Identify the measurement of angle \( ABC \). Given that \( \angle ABC = 68^\circ \), we can find the measurement of \( \angle ABD \) and \( \angle DBC \).
2. **Step 2:** Since ray \( BD \) is the angle bisector of \( \angle ABC \), it divides it into two equal angles:
\[
\angle ABD = \angle DBC = \frac{1}{2} \times 68^\circ = 34^\circ
\]
3. **Step 3:** Now, we have \( \angle DBC = 34^\circ \). Next, we focus on \( \angle DBC \) where ray \( BE \) is an angle bisector that divides \( \angle DBC \) into two equal angles (\( \angle EBD \) and \( \angle EBC \)).
4. **Step 4:** Again, applying the property of the angle bisector:
\[
\angle EBD = \angle EBC = \frac{1}{2} \times 34^\circ = 17^\circ
\]
Therefore, the measurement of angle \( EBC \) is \( 17^\circ \).
1. **Step 1:** Identify the measurement of angle \( ABC \). Given that \( \angle ABC = 68^\circ \), we can find the measurement of \( \angle ABD \) and \( \angle DBC \).
2. **Step 2:** Since ray \( BD \) is the angle bisector of \( \angle ABC \), it divides it into two equal angles:
\[
\angle ABD = \angle DBC = \frac{1}{2} \times 68^\circ = 34^\circ
\]
3. **Step 3:** Now, we have \( \angle DBC = 34^\circ \). Next, we focus on \( \angle DBC \) where ray \( BE \) is an angle bisector that divides \( \angle DBC \) into two equal angles (\( \angle EBD \) and \( \angle EBC \)).
4. **Step 4:** Again, applying the property of the angle bisector:
\[
\angle EBD = \angle EBC = \frac{1}{2} \times 34^\circ = 17^\circ
\]
Therefore, the measurement of angle \( EBC \) is \( 17^\circ \).
Answered by
GPT-4o mini
To determine which set of side lengths does not form a triangle, we can use the triangle inequality theorem. The triangle inequality states that for three lengths \(a\), \(b\), and \(c\), they can form a triangle if and only if the following conditions hold:
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)
Let's check each set of side lengths:
1. **Set: 2, 3, 5**
- \(2 + 3 = 5\) (not greater than 5)
- \(2 + 5 = 7\) (greater than 3)
- \(3 + 5 = 8\) (greater than 2)
- **This set does NOT satisfy the triangle inequality.**
2. **Set: 3, 5, 7**
- \(3 + 5 = 8\) (greater than 7)
- \(3 + 7 = 10\) (greater than 5)
- \(5 + 7 = 12\) (greater than 3)
- **This set satisfies the triangle inequality.**
3. **Set: 5, 7, 10**
- \(5 + 7 = 12\) (greater than 10)
- \(5 + 10 = 15\) (greater than 7)
- \(7 + 10 = 17\) (greater than 5)
- **This set satisfies the triangle inequality.**
4. **Set: 6, 14, 18**
- \(6 + 14 = 20\) (greater than 18)
- \(6 + 18 = 24\) (greater than 14)
- \(14 + 18 = 32\) (greater than 6)
- **This set satisfies the triangle inequality.**
**Conclusion:** The set of side lengths **2, 3, 5** does not form a triangle.
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)
Let's check each set of side lengths:
1. **Set: 2, 3, 5**
- \(2 + 3 = 5\) (not greater than 5)
- \(2 + 5 = 7\) (greater than 3)
- \(3 + 5 = 8\) (greater than 2)
- **This set does NOT satisfy the triangle inequality.**
2. **Set: 3, 5, 7**
- \(3 + 5 = 8\) (greater than 7)
- \(3 + 7 = 10\) (greater than 5)
- \(5 + 7 = 12\) (greater than 3)
- **This set satisfies the triangle inequality.**
3. **Set: 5, 7, 10**
- \(5 + 7 = 12\) (greater than 10)
- \(5 + 10 = 15\) (greater than 7)
- \(7 + 10 = 17\) (greater than 5)
- **This set satisfies the triangle inequality.**
4. **Set: 6, 14, 18**
- \(6 + 14 = 20\) (greater than 18)
- \(6 + 18 = 24\) (greater than 14)
- \(14 + 18 = 32\) (greater than 6)
- **This set satisfies the triangle inequality.**
**Conclusion:** The set of side lengths **2, 3, 5** does not form a triangle.
Answered by
GPT-4o mini
To construct a copy of line segment \( AB \) using point \( C \) as one of the endpoints, Carrie can follow these steps:
1. **Step 1:** Measure the length of segment \( AB \). Carrie will use her compass for this. She should place the compass point at point \( A \) and adjust the compass width to reach point \( B \) so that the distance between points is represented in the compass.
2. **Step 2:** Without changing the width of the compass, Carrie should then move to point \( C \). She can place the point of the compass on point \( C \).
3. **Step 3:** With the compass point fixed at \( C \), Carrie should swing an arc that intersects the ray extending from \( C \) (in any direction) to create a new point \( D \). The length \( CD \) will be equal to the length \( AB \).
4. **Step 4:** Finally, Carrie can draw line segment \( CD \) to complete the construction of the copy of segment \( AB \).
So, the actual step she needs to take next is to **set the compass to the length of segment \( AB \)** by placing the compass point at \( A \) and marking point \( B \), and then use that length to draw an arc from point \( C \).
1. **Step 1:** Measure the length of segment \( AB \). Carrie will use her compass for this. She should place the compass point at point \( A \) and adjust the compass width to reach point \( B \) so that the distance between points is represented in the compass.
2. **Step 2:** Without changing the width of the compass, Carrie should then move to point \( C \). She can place the point of the compass on point \( C \).
3. **Step 3:** With the compass point fixed at \( C \), Carrie should swing an arc that intersects the ray extending from \( C \) (in any direction) to create a new point \( D \). The length \( CD \) will be equal to the length \( AB \).
4. **Step 4:** Finally, Carrie can draw line segment \( CD \) to complete the construction of the copy of segment \( AB \).
So, the actual step she needs to take next is to **set the compass to the length of segment \( AB \)** by placing the compass point at \( A \) and marking point \( B \), and then use that length to draw an arc from point \( C \).
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