Solved on Nov 21, 2023

Graph a triangle with vertices $A(3,1), B(3,4), C(7,1)$ . Classify the triangle by its sides and determine if it is a right triangle.

STEP 1

Assumptions1. The vertices of the triangle are A(3,1), B(3,4), and C(7,1). . We are working in a two-dimensional Cartesian coordinate system.
3. The classification of the triangle is based on the lengths of its sides.
4. To determine if it's a right triangle, we will use the Pythagorean theorem.

STEP 2

First, we need to plot the points A(,1), B(,4), and C(7,1) on the Cartesian plane.

STEP 3

Next, we connect the points to form the triangle ABC.

STEP 4

Now, we need to calculate the lengths of the sides of the triangle. We can do this using the distance formula $d = \sqrt{(x2-x1)^2 + (y2-y1)^2}$

STEP 5

Let's calculate the length of side AB.
$AB = \sqrt{(3-3)^2 + (4-1)^2}$

STEP 6

implify the expression to find the length of side AB.
$AB = \sqrt{0 +3^2} = \sqrt{9} =3$

STEP 7

Now, let's calculate the length of side BC.
$BC = \sqrt{(7-3)^2 + (1-4)^2}$

STEP 8

implify the expression to find the length of side BC.
$BC = \sqrt{4^2 + (-3)^2} = \sqrt{16 +} = \sqrt{25} =5$

STEP 9

Finally, let's calculate the length of side AC.
$AC = \sqrt{(7-3)^2 + (-)^2}$

STEP 10

implify the expression to find the length of side AC.
$AC = \sqrt{4^2 +0} = \sqrt{16} =4$

STEP 11

Now that we have the lengths of all sides, we can classify the triangle by its sides. Since all sides have different lengths, the triangle is scalene.

STEP 12

To determine if the triangle is a right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. We can write this as $c^2 = a^2 + b^2$ where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

STEP 13

Let's check if the Pythagorean theorem holds for our triangle. We'll use the longest side (BC) as the hypotenuse.
$5^2 =3^2 +^2$

STEP 14

implify the expression to check if the Pythagorean theorem holds.
$25 =9 +16 =25$ Since the Pythagorean theorem holds for our triangle, it is a right triangle.
So, the triangle ABC with vertices A(3,), B(3,4), and C(7,) is a scalene right triangle.

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Graph a triangle with vertices $A(3,1), B(3,4), C(7,1)$ . Classify the triangle by its sides and determine if it is a right triangle.

STEP 1

Assumptions1. The vertices of the triangle are A(3,1), B(3,4), and C(7,1). . We are working in a two-dimensional Cartesian coordinate system.
3. The classification of the triangle is based on the lengths of its sides.
4. To determine if it's a right triangle, we will use the Pythagorean theorem.

STEP 2

First, we need to plot the points A(,1), B(,4), and C(7,1) on the Cartesian plane.

STEP 3

Next, we connect the points to form the triangle ABC.

STEP 4

Now, we need to calculate the lengths of the sides of the triangle. We can do this using the distance formula $d = \sqrt{(x2-x1)^2 + (y2-y1)^2}$

STEP 5

Let's calculate the length of side AB.
$AB = \sqrt{(3-3)^2 + (4-1)^2}$

STEP 6

implify the expression to find the length of side AB.
$AB = \sqrt{0 +3^2} = \sqrt{9} =3$

STEP 7

Now, let's calculate the length of side BC.
$BC = \sqrt{(7-3)^2 + (1-4)^2}$

STEP 8

implify the expression to find the length of side BC.
$BC = \sqrt{4^2 + (-3)^2} = \sqrt{16 +} = \sqrt{25} =5$

STEP 9

Finally, let's calculate the length of side AC.
$AC = \sqrt{(7-3)^2 + (-)^2}$

STEP 10

implify the expression to find the length of side AC.
$AC = \sqrt{4^2 +0} = \sqrt{16} =4$

STEP 11

Now that we have the lengths of all sides, we can classify the triangle by its sides. Since all sides have different lengths, the triangle is scalene.

STEP 12

STEP 13

Let's check if the Pythagorean theorem holds for our triangle. We'll use the longest side (BC) as the hypotenuse.
$5^2 =3^2 +^2$

STEP 14

implify the expression to check if the Pythagorean theorem holds.
$25 =9 +16 =25$ Since the Pythagorean theorem holds for our triangle, it is a right triangle.
So, the triangle ABC with vertices A(3,), B(3,4), and C(7,) is a scalene right triangle.

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Graph a triangle with vertices A(3,1),B(3,4),C(7,1)A(3,1), B(3,4), C(7,1)A(3,1),B(3,4),C(7,1). Classify the triangle by its sides and determine if it is a right triangle.

STEP 1

Assumptions1. The vertices of the triangle are A(3,1), B(3,4), and C(7,1). . We are working in a two-dimensional Cartesian coordinate system.3. The classification of the triangle is based on the lengths of its sides.4. To determine if it's a right triangle, we will use the Pythagorean theorem.

STEP 2

First, we need to plot the points A(,1), B(,4), and C(7,1) on the Cartesian plane.

STEP 3

Next, we connect the points to form the triangle ABC.

STEP 4

Now, we need to calculate the lengths of the sides of the triangle. We can do this using the distance formulad=(x2−x1)2+(y2−y1)2d = \sqrt{(x2-x1)^2 + (y2-y1)^2}d=(x2−x1)2+(y2−y1)2​

STEP 5

Let's calculate the length of side AB.AB=(3−3)2+(4−1)2AB = \sqrt{(3-3)^2 + (4-1)^2}AB=(3−3)2+(4−1)2​

STEP 6

implify the expression to find the length of side AB.AB=0+32=9=3AB = \sqrt{0 +3^2} = \sqrt{9} =3AB=0+32​=9​=3

STEP 7

Now, let's calculate the length of side BC.BC=(7−3)2+(1−4)2BC = \sqrt{(7-3)^2 + (1-4)^2}BC=(7−3)2+(1−4)2​

STEP 8

implify the expression to find the length of side BC.BC=42+(−3)2=16+=25=5BC = \sqrt{4^2 + (-3)^2} = \sqrt{16 +} = \sqrt{25} =5BC=42+(−3)2​=16+​=25​=5

STEP 9

Finally, let's calculate the length of side AC.AC=(7−3)2+(−)2AC = \sqrt{(7-3)^2 + (-)^2}AC=(7−3)2+(−)2​

STEP 10

implify the expression to find the length of side AC.AC=42+0=16=4AC = \sqrt{4^2 +0} = \sqrt{16} =4AC=42+0​=16​=4

STEP 11

Now that we have the lengths of all sides, we can classify the triangle by its sides. Since all sides have different lengths, the triangle is scalene.

STEP 12

STEP 13

Let's check if the Pythagorean theorem holds for our triangle. We'll use the longest side (BC) as the hypotenuse.52=32+25^2 =3^2 +^252=32+2

STEP 14

implify the expression to check if the Pythagorean theorem holds.25=9+16=2525 =9 +16 =2525=9+16=25Since the Pythagorean theorem holds for our triangle, it is a right triangle.So, the triangle ABC with vertices A(3,), B(3,4), and C(7,) is a scalene right triangle.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Graph a triangle with vertices $A(3,1), B(3,4), C(7,1)$ . Classify the triangle by its sides and determine if it is a right triangle.

Assumptions1. The vertices of the triangle are A(3,1), B(3,4), and C(7,1). . We are working in a two-dimensional Cartesian coordinate system.
3. The classification of the triangle is based on the lengths of its sides.
4. To determine if it's a right triangle, we will use the Pythagorean theorem.

Now, we need to calculate the lengths of the sides of the triangle. We can do this using the distance formula $d = \sqrt{(x2-x1)^2 + (y2-y1)^2}$

Let's calculate the length of side AB.
$AB = \sqrt{(3-3)^2 + (4-1)^2}$

implify the expression to find the length of side AB.
$AB = \sqrt{0 +3^2} = \sqrt{9} =3$

Now, let's calculate the length of side BC.
$BC = \sqrt{(7-3)^2 + (1-4)^2}$

implify the expression to find the length of side BC.
$BC = \sqrt{4^2 + (-3)^2} = \sqrt{16 +} = \sqrt{25} =5$

Finally, let's calculate the length of side AC.
$AC = \sqrt{(7-3)^2 + (-)^2}$

implify the expression to find the length of side AC.
$AC = \sqrt{4^2 +0} = \sqrt{16} =4$

Let's check if the Pythagorean theorem holds for our triangle. We'll use the longest side (BC) as the hypotenuse.
$5^2 =3^2 +^2$

implify the expression to check if the Pythagorean theorem holds.
$25 =9 +16 =25$ Since the Pythagorean theorem holds for our triangle, it is a right triangle.
So, the triangle ABC with vertices A(3,), B(3,4), and C(7,) is a scalene right triangle.