Math  /  Geometry

Question2.) Triangle ABCA B C has angle A=70A=70 degrees, side c=26 cmc=26 \mathrm{~cm} and side a=25 cma=25 \mathrm{~cm}. Draw the triangle and find angle BB and CC as well as side bb.

Studdy Solution

STEP 1

1. We are given a triangle ABC \triangle ABC .
2. Angle A A is 70 70^\circ .
3. Side c c (opposite angle C C ) is 26 26 cm.
4. Side a a (opposite angle A A ) is 25 25 cm.
5. We need to find angles B B and C C , and side b b (opposite angle B B ).

STEP 2

1. Draw the triangle with the given information.
2. Use the Law of Sines to find angle C C .
3. Calculate angle B B using the angle sum property of triangles.
4. Use the Law of Sines to find side b b .

STEP 3

Draw a triangle ABC \triangle ABC with angle A=70 A = 70^\circ , side a=25 a = 25 cm, and side c=26 c = 26 cm. Label the vertices and sides accordingly.

STEP 4

Use the Law of Sines to find angle C C . The Law of Sines states:
asinA=csinC\frac{a}{\sin A} = \frac{c}{\sin C}
Substitute the known values:
25sin70=26sinC\frac{25}{\sin 70^\circ} = \frac{26}{\sin C}

STEP 5

Solve for sinC\sin C:
sinC=26sin7025\sin C = \frac{26 \cdot \sin 70^\circ}{25}
Calculate sin70\sin 70^\circ and then solve for sinC\sin C.

STEP 6

Assuming sin700.9397\sin 70^\circ \approx 0.9397, calculate sinC\sin C:
sinC=260.9397250.9773\sin C = \frac{26 \cdot 0.9397}{25} \approx 0.9773
Find angle C C using sin1\sin^{-1}:
Csin1(0.9773)77.4C \approx \sin^{-1}(0.9773) \approx 77.4^\circ

STEP 7

Use the angle sum property of triangles to find angle B B :
A+B+C=180A + B + C = 180^\circ
Substitute the known angles:
70+B+77.4=18070^\circ + B + 77.4^\circ = 180^\circ
Solve for B B :
B=1807077.432.6B = 180^\circ - 70^\circ - 77.4^\circ \approx 32.6^\circ

STEP 8

Use the Law of Sines to find side b b :
bsinB=asinA\frac{b}{\sin B} = \frac{a}{\sin A}
Substitute the known values:
bsin32.6=25sin70\frac{b}{\sin 32.6^\circ} = \frac{25}{\sin 70^\circ}

STEP 9

Solve for b b :
b=25sin32.6sin70b = \frac{25 \cdot \sin 32.6^\circ}{\sin 70^\circ}
Assuming sin32.60.5395\sin 32.6^\circ \approx 0.5395, calculate b b :
b250.53950.939714.35 cmb \approx \frac{25 \cdot 0.5395}{0.9397} \approx 14.35 \text{ cm}
The angles B B and C C are approximately 32.6 32.6^\circ and 77.4 77.4^\circ , respectively. The side b b is approximately 14.35 14.35 cm.

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