Answer:
Step-by-step explanation:
11 is incomplete, can't solve
12
The triangle is equilateral, so all sides are equal, using one pair to find x:
Each side is:
13.
Sides are equal as triangle is equilateral
Finding x by comparing two sides
Sides are equal
14.
Equal sides of isosceles triangle:
Sides are
15.
Equal sides of isosceles triangle WXY, WX = WY
Sides are:
11. Applying distance formula and finding the sides of DEF, triangle DEF can be classified as an: isosceles triangle.
Applying the definition of equilateral and isosceles triangles, the value of x and the measure of the sides of the given triangles are as follows:
12. x = 6
JK = KL = JL = 83
13. x = 18
QR = RS = QS = 29
14. x = 11
DE = CD = 74
CE = 37
15. x = 23
WY = WX = 95
XY = 108
Recall:
11. Given the vertices:
Find DE, DF, and EF using [tex]d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex]
Length of DE:
[tex]DE = \sqrt{(5 -(-2))^2 + (5 - 3)^2} \\\\DE = \sqrt{(7)^2 + (2)^2} \\\\\mathbf{DE = \sqrt{53}}[/tex]
Length of DF:
[tex]DF = \sqrt{(-4 -(-2))^2 + (10 - 3)^2} \\\\DF = \sqrt{(-2)^2 + (7)^2} \\\\\mathbf{DF = \sqrt{53}}[/tex]
Length of EF:
[tex]EF = \sqrt{(-4 -5)^2 + (10 - 5)^2} \\\\EF = \sqrt{(-9)^2 + (5)^2} \\\\\mathbf{EF = \sqrt{106}}[/tex]
Triangle DEF has two equal sides, therefore, it is an isosceles triangle.
12. Given that triangle JKL is equilateral, all it's side will be equal.
Therefore, JK = JL
[tex]13x + 5 = 8x + 35[/tex]
[tex]13x + 5 = 8x + 35\\\\13x - 8x = -5 + 35\\\\5x = 30\\\\\mathbf{x = 6}[/tex]
Plug in the value of x to find the measure of the sides of triangle JKL.
[tex]JK = 13x + 5 = 13(6) + 5\\\\\mathbf{JK = 83}[/tex]
[tex]KL = 17x - 19 = 17(6) - 19\\\\\mathbf{KL = 83}[/tex]
[tex]JL = 8x + 35 = 8(6) + 35\\\\\mathbf{JL = 83}[/tex]
JK = KL = JL = 83
13. Triangle QRS is equilateral, meaning all its side are equal.
To find x, set QR equal to RS (equal sides).
[tex]2x - 7 = 5x - 61[/tex]
[tex]2x - 5x = 7 - 61\\\\-3x = -54\\\\\mathbf{x = 18}[/tex]
Plug in the value of x to find the measure of the sides of triangle QRS.
[tex]QR = 2x - 7 = 2(18) - 7\\\\\mathbf{QR = 29}[/tex]
14. Since CDE is isosceles, therefore, CD = DE.
[tex]9x - 25 = 6x + 8\\\\9x - 6x = 25 + 8\\\\3x = 33\\\\\mathbf{x = 11}[/tex]
Plug in the value of x to find the measure of the sides of triangle CDE.
[tex]CD = 9x - 25 = 9(11) - 25\\\\\mathbf{CD = 74}[/tex]
DE = CD = 74 (congruent sides)
[tex]CE = 10x - 73 = 10(11) - 73\\\\\mathbf{CE = 37}[/tex]
15. We are given that triangle WXY is isosceles, therefore, WX = WY.
[tex]4x + 3 = 7x - 66\\\\4x - 7x = -3 - 66\\\\-3x = -69\\\\\mathbf{x = 23}[/tex]
Plug in the value of x to find the measure of the sides of triangle WXY.
[tex]WX = 4x + 3 = 4(23) + 3\\\\\mathbf{WX = 95}[/tex]
WY = WX = 95 (congruent sides).
[tex]XY = 5x - 7 = 5(23) - 7\\\\\mathbf{XY = 108}[/tex]
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