Question

solve for ( x ). round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.
(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 ))
answer attempt 1 out of 2
( x = \boxed{29.0} ) submit 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:

\(29.0\)