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In triangle xyz xy equals 6cm yz equals 7cm angle y equals 78 degrees find the length of xz and the measure of angle x give answers to one decimal place?
Wiki User
∙ 14y ago
I know there is a equation to find the answer, but if youae like me someday you will forget it. So, I am going to use trig functions and a little work.
Derive the side and angle of a triangle
In triangle xyz, yx equals 6cm, yz equals 7cm, angle y equals 78 degrees find the length of xz and the measure of angle x give answers to one decimal place?
I know there is an equation to find the answer, but if you are like me, someday you will forget it. So, I am going to use trig functions and a little work.
I wish I wiki would copy MS word drawings, but here we go.
Draw a triangle with the base, YX = 6 cm.
Label the left end of the line as Point Y.
Starting at Point Y, draw a 7 cm line (YZ) up and right from the base YX making an angle of 78º angle with the base.
Label the end point of this line as Point Z, so this is line YZ
The angle XYZ should open to the right. The only reason I keep repeating this is so your picture looks like mine and you can follow my directions.
Connect the points X and Z to complete the triangle.
Draw an altitude straight down from point Z.
Label the point where the altitude meets the base as point A.
The altitude is line ZA
Now you have 2 right triangles.
Look at triangle YZA (left one)
Label the 78º angle.
Sin θ = opposite ÷ hypotenuse
Sin 78º = ZA ÷ YZ
Sin 78º = ZA ÷ 7
ZA = 6.85 cm (Altitude)
Cos θ = adjacent ÷ hypotenuse
Hypotenuse = YZ = 7 cm
Adjacent = line YA
Cos 78º = YA ÷ 7
YA = 7 * Cos 78º
YA = 1.46 cm
YX = 6 cm
YX = YA + AX
6 = 1.46 + AX
AX = 4.54 cm
Now we know the length of 2 sides of the right triangle on the right side, right Triangle ZXA.
ZA = 6.85 cm (Altitude)
AX = 4.54 cm (adjacent)
Tan θ (ZXA) = opposite ÷ adjacent
Opposite = ZA (altitude) = 6.85 cm
Adjacent= AX = 4.54 cm
Tan θ (ZXA) = 6.85 ÷ 4.54 cm
(ZXA) θ =56.5º
In the right Triangle ZXA, line ZX is the hypotenuse, the line ZA(Altitude) is the opposite, and the angle between these two lines = 56.5º.
Sin θ = opposite ÷ hypotenuse
Sin 56.5º = Altitude ÷ hypotenuse
Sin 56.5º = 6.85 ÷ hypotenuse
Hypotenuse = 6.85 ÷ Sin 56.5º
Hypotenuse =8.22 cm
ZX = 8.22 cm
find the length of xz and the measure of angle x give answers to one decimal place?
ZX = 8.22 cm, θ =56.5º
Here is the work from the beginning to the end.
ZX = (7 * sin 78º) ÷ (7 * Cos 78º - 6) *-1, tan-1, sin, 1/x, * (7* sin 78º)
Wiki User
∙ 14y ago
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