Question

Triangles ABE, ADE, and CBE are shown on the coordinate grid, and all the vertices have coordinates that are integers. Which statement is true?

Answer 1

To check if Triangles ABE, ADE, and CBE are congruent, let us compute for the distance of each line using the Distance Formula,

`\text{ }d\text{ = }\sqrt[]{(x_2-x_1)^2\text{ + (}y_2-y_1)^2}`

Where,

d = Distance

(x1, y1) = Coordinates of the first point

(x2, y2) = Coordinates of the second point

Let's compute the distance of the following lines:

Triangle ABE: Lines AB, AE, and BE

Triangle ADE: Lines AD, AE, and ED

Triangle CBE: Lines CE, CB, and BE

For Triangle ABE,

`\text{ d}_(AB)\text{ = }\sqrt[]{(-1-(-4))^2+(3-(-1))^2}\text{ = }\sqrt[]{(-1+4)^2+(3+1)^2}`
`\text{ d}_(AB)\text{ = }\sqrt[]{(3)^2+(4)^2}\text{ = }\sqrt[]{9+\text{ 16}}\text{ = }\sqrt[]{25}`
`\text{ d}_(AB)\text{ = 5}`
`\text{ d}_(AE)\text{ =}\sqrt[]{(1-\text{ }(-1))^2+(0\text{ - }(-4))^2}\text{ = }\sqrt[]{(1+1)^2+(0+4)^2}`
`\text{ d}_(AE)\text{ = }\sqrt[]{(2)^2+(4)^2}\text{ = }\sqrt[]{4\text{ + 16}}`
`\text{ d}_(AE)\text{ =}\sqrt[]{20}`
`\text{ d}_(BE)\text{ = }\sqrt[]{(1\text{ - (}3))^2+(0\text{ - (-1)})^2}\text{ = }\sqrt[]{(1-3)^2+(0+1)^2}`
`\text{ d}_(BE)=\text{ }\sqrt[]{(-2)^2+(1)^2}\text{ = }\sqrt[]{4\text{ + 1}}`
`\text{ d}_(BE)\text{ = }\sqrt[]{5}`

For Triangle ADE, let's compute for the distance of line AD and ED since we already got the distance of line AE.

`\text{ d}_(AD)\text{ = }\sqrt[]{(-1-(-1))^2+\text{ (}1\text{ - }(-4))^2}\text{ = }\sqrt[]{(-1+1)^2+(1+4)^2}`
`\text{ d}_(AD)\text{ = }\sqrt[]{(0)^2+(5)^2}\text{ = }\sqrt[]{25}`
`\text{ d}_(AD)\text{ = 5}`
`\text{ d}_(ED)=\text{ }\sqrt[]{(-1\text{ - (}1))^2+(1-0)^2}\text{ = }\sqrt[]{(-1-1)^2+(1)^2}`
`\text{ d}_(ED)\text{ = }\sqrt[]{(-2)^2_{}+(1)^2}\text{ = }\sqrt[]{4\text{ + 1}}`
`\text{ d}_(ED)\text{ = }\sqrt[]{5}`

For Triangle CBE, let's compute for the distance of line CE and CB since we already got the distance of line BE.

`\text{ d}_(CE)\text{ = }\sqrt[]{(3-\text{ }1)^2+(4-0)^2}\text{ = }\sqrt[]{(2)^2+(4)^2}`
`\text{ d}_(CE)\text{ = }\sqrt[]{4+16}\text{ = }\sqrt[]{20}`
`\text{ d}_(CE)\text{ = }\sqrt[]{20}`
`\text{ d}_(CB)\text{ = }\sqrt[]{(3-3)^2+(4\text{ - (}-1))^2}\text{ =}\sqrt[]{(0)^2+(4+1)^2}`
`\text{ d}_(CB)\text{ =}\sqrt[]{(5)^2}\text{ = }\sqrt[]{25}`
`\text{ d}_(CB)\text{ = 5}`

In summary,

Triangle ABE:

`AB=\text{ 5, AE = }\sqrt[]{20}\text{ and BE = }\sqrt[]{5}`

Triangle ADE:

`\text{ AD = 5, AE = }\sqrt[]{20}\text{ and ED = }\sqrt[]{5}`

Triangle CBE: CE, CB, and BE

`\text{ CB = 5, CE = }\sqrt[]{20}\text{ and BE = }\sqrt[]{5}`

The sides of the three triangles shown in the grid are congruent based on the SSS Rule of Triangle.

Thus, the statement that meets our evaluation is:

D. Triangle ABE, ADE and CBE are all congruent.