Question

technology required. abc, shown here, is a right triangle.
find the unknown side length and angle measures.
type your answers in the boxes.
bc =
units
measure of ∠c =
°
measure of ∠b =
°

Explanation:

Step1: Find BC using Pythagorean theorem

In right triangle \(ABC\), \(AB = 21\), \(AC = 35\). By Pythagorean theorem, \(BC=\sqrt{AB^{2}+AC^{2}}=\sqrt{21^{2}+35^{2}}=\sqrt{441 + 1225}=\sqrt{1666}\approx40.82\) (Wait, actually, \(21 = 7\times3\), \(35=7\times5\), so \(BC = 7\sqrt{3^{2}+5^{2}}=7\sqrt{34}\approx7\times5.830\approx40.81\)? Wait, no, \(21^2=441\), \(35^2 = 1225\), sum is \(441+1225 = 1666\)? Wait, no, \(21^2=441\), \(35^2=1225\), 441+1225=1666? Wait, 2121=441, 3535=1225, 441+1225=1666. Then \(\sqrt{1666}\approx40.82\). Wait, but maybe I made a mistake. Wait, 21 and 35, GCD is 7, so 21=73, 35=75. Then \(BC = 7\sqrt{3^2 + 5^2}=7\sqrt{9 + 25}=7\sqrt{34}\approx7\times5.83095\approx40.82\).

Step2: Find angle C using tangent

\(\tan(C)=\frac{AB}{AC}=\frac{21}{35}=\frac{3}{5}=0.6\). So \(C=\arctan(0.6)\approx30.96^\circ\)

Step3: Find angle B using tangent or complementary angles

Since it's a right triangle, \(A = 90^\circ\), so \(B + C=90^\circ\). So \(B = 90^\circ - C\approx90 - 30.96 = 59.04^\circ\). Alternatively, \(\tan(B)=\frac{AC}{AB}=\frac{35}{21}=\frac{5}{3}\approx1.6667\), so \(B=\arctan(\frac{5}{3})\approx59.04^\circ\)

Wait, let's recalculate BC: \(AB = 21\), \(AC = 35\). So \(BC=\sqrt{21^2 + 35^2}=\sqrt{441 + 1225}=\sqrt{1666}\approx40.82\). Let's check with calculator: 21 squared is 441, 35 squared is 1225, sum is 1666. Square root of 1666: 40^2=1600, 41^2=1681, so 40.82^2≈1666.

For angle C: opposite side is AB=21, adjacent is AC=35. So \(\tan(C)=\frac{21}{35}=0.6\), so \(C=\arctan(0.6)\approx30.96^\circ\), which is approximately 31 degrees.

Angle B: opposite side AC=35, adjacent AB=21. \(\tan(B)=\frac{35}{21}=\frac{5}{3}\approx1.6667\), so \(B=\arctan(\frac{5}{3})\approx59.04^\circ\), approximately 59 degrees.

Wait, maybe the problem expects exact values or rounded to one decimal. Let's do more precise calculations.

\(\arctan(0.6)\): using calculator, 0.6, arctan is approximately 30.96375653 degrees, so ~31.0 degrees.

\(\arctan(\frac{5}{3})\): 5/3≈1.6667, arctan is approximately 59.03624347 degrees, ~59.0 degrees.

And BC: \(\sqrt{21^2 + 35^2}=\sqrt{441 + 1225}=\sqrt{1666}\approx40.82\) (or 7√34≈40.817, which is ~40.82)

Answer:

\(BC\approx\boxed{40.82}\) units
Measure of \(\angle C\approx\boxed{30.96}\)° (or 31°)
Measure of \(\angle B\approx\boxed{59.04}\)° (or 59°)

(Note: Depending on rounding, the answers can be rounded to whole numbers or one decimal place. For example, if rounded to the nearest whole number: BC≈41, ∠C≈31°, ∠B≈59°)