solve for ( x ). round your answer to the nearest tenth if necessary. figures are not necessarily drawn to…

solve for ( x ). round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.\n(there are two triangles in the image, one with vertices u, t, s and sides ut=23, us=24, angles at t is 72°, at s is 63°; the other with vertices w, v, x and sides xv=31, angle at x is 45°, at v is 63°, and side wx is ( x ))\nanswer attempt 1 out of 2\n( x = \boxed{29.0} ) submit answer

solve for ( x ). round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.\n(there are two triangles in the image, one with vertices u, t, s and sides ut=23, us=24, angles at t is 72°, at s is 63°; the other with vertices w, v, x and sides xv=31, angle at x is 45°, at v is 63°, and side wx is ( x ))\nanswer attempt 1 out of 2\n( x = \boxed{29.0} ) submit answer

Answer

Explanation:

Step1: Find the third angle of triangle UST

In triangle UST, the sum of angles is (180^\circ). So the third angle at U is (180^\circ - 72^\circ - 63^\circ = 45^\circ). Now we see that triangle UST and triangle XWV have two angles equal: angle at S ((63^\circ)) equals angle at V ((63^\circ)), and angle at U ((45^\circ)) equals angle at X ((45^\circ)). So the triangles are similar by AA (Angle - Angle) similarity criterion.

Step2: Set up the proportion for similar triangles

For similar triangles, the ratios of corresponding sides are equal. In triangle UST, side opposite (63^\circ) (angle at T) is (24), and side opposite (45^\circ) (angle at U) is (23)? Wait, no, let's match the angles. Triangle UST: angles (45^\circ) (U), (63^\circ) (S), (72^\circ) (T). Triangle XWV: angles (45^\circ) (X), (63^\circ) (V), so the third angle at W is (180 - 45 - 63 = 72^\circ). So corresponding sides: in UST, side (US = 24) (opposite (72^\circ) at T), (ST) (opposite (45^\circ) at U) is? Wait, maybe better to match the angles. Angle (45^\circ) (U in UST, X in XWV), angle (63^\circ) (S in UST, V in XWV). So side (US) (in UST, between (45^\circ) and (63^\circ)) corresponds to side (XV) (in XWV, between (45^\circ) and (63^\circ)) which is (31). Side (ST) (in UST, between (63^\circ) and (72^\circ))? Wait, no, let's use the Law of Sines. In triangle UST: (\frac{US}{\sin 72^\circ}=\frac{ST}{\sin 45^\circ}=\frac{UT}{\sin 63^\circ}). In triangle XWV: (\frac{XW}{\sin 63^\circ}=\frac{WV}{\sin 45^\circ}=\frac{XV}{\sin 72^\circ}). Since (XV = 31) (side in XWV, opposite (72^\circ) at W), and (US = 24) (side in UST, opposite (72^\circ) at T). Wait, maybe the correct correspondence is: angle (45^\circ) (U) - angle (45^\circ) (X), angle (63^\circ) (S) - angle (63^\circ) (V), so side (UT = 23) (in UST, between (45^\circ) and (72^\circ)) corresponds to side (WV) (in XWV, between (45^\circ) and (72^\circ))? No, let's use Law of Sines on both triangles.

In triangle UST: (\frac{UT}{\sin 63^\circ}=\frac{US}{\sin 72^\circ}=\frac{ST}{\sin 45^\circ})

In triangle XWV: (\frac{XW}{\sin 63^\circ}=\frac{WV}{\sin 45^\circ}=\frac{XV}{\sin 72^\circ})

We know (XV = 31), (US = 24). From triangle UST, (\frac{US}{\sin 72^\circ}=\frac{24}{\sin 72^\circ}). From triangle XWV, (\frac{XV}{\sin 72^\circ}=\frac{31}{\sin 72^\circ}). Wait, maybe the sides: in UST, side (UT = 23) (opposite (63^\circ) at S), side (US = 24) (opposite (72^\circ) at T), side (ST) (opposite (45^\circ) at U). In XWV, side (XW = x) (opposite (63^\circ) at V), side (XV = 31) (opposite (72^\circ) at W), side (WV) (opposite (45^\circ) at X). So by Law of Sines, in UST: (\frac{UT}{\sin 63^\circ}=\frac{US}{\sin 72^\circ}), so (\frac{23}{\sin 63^\circ}=\frac{24}{\sin 72^\circ})? Wait, no, that should hold for similar triangles. Wait, maybe I made a mistake in angle correspondence. Let's recalculate the angles.

Triangle UST: angles (45^\circ) (U), (63^\circ) (S), (72^\circ) (T). So sides: (UT) is opposite (63^\circ) (S), (US) is opposite (72^\circ) (T), (ST) is opposite (45^\circ) (U).

Triangle XWV: angles (45^\circ) (X), (63^\circ) (V), (72^\circ) (W). So sides: (XW) is opposite (63^\circ) (V), (WV) is opposite (45^\circ) (X), (XV) is opposite (72^\circ) (W).

So corresponding sides: (UT) (opposite (63^\circ) in UST) corresponds to (XW) (opposite (63^\circ) in XWV), (US) (opposite (72^\circ) in UST) corresponds to (XV) (opposite (72^\circ) in XWV), (ST) (opposite (45^\circ) in UST) corresponds to (WV) (opposite (45^\circ) in XWV).

So by Law of Sines, (\frac{UT}{XW}=\frac{US}{XV})

We know (UT = 23), (US = 24), (XV = 31). Wait, no, (\frac{US}{XV}=\frac{24}{31}), and (\frac{UT}{XW}=\frac{23}{x}). But since triangles are similar, (\frac{US}{XV}=\frac{UT}{XW})? Wait, no, maybe (\frac{US}{XV}=\frac{ST}{WV}=\frac{UT}{XW}). Wait, let's use Law of Sines on triangle UST: (\frac{UT}{\sin 63^\circ}=\frac{US}{\sin 72^\circ})

So (\frac{23}{\sin 63^\circ}=\frac{24}{\sin 72^\circ})? Let's check: (\sin 63^\circ\approx 0.891), (\sin 72^\circ\approx 0.951). (\frac{23}{0.891}\approx 25.8), (\frac{24}{0.951}\approx 25.2). Close, maybe due to rounding. Now in triangle XWV, Law of Sines: (\frac{XW}{\sin 63^\circ}=\frac{XV}{\sin 72^\circ})

So (x=\frac{31\times\sin 63^\circ}{\sin 72^\circ})

Calculate (\sin 63^\circ\approx 0.891), (\sin 72^\circ\approx 0.951)

(x=\frac{31\times 0.891}{0.951}\approx\frac{27.621}{0.951}\approx 29.0)

Answer:

(29.0)