Victory for the ACT Exam 16e TG Sample

U NIT 4 | G EOMETRY • 509

STUDENT TEXT, pp. 185–186 9. I o n f the triangle below, what is the length AC ?

EXPLANATIONS

9. (C) Mathematics/Geometry/

30°-60°-90° Triangles Item #9

Triangles/30° − 60° − 90° Triangles CC: HSG-SRT.C.6 CCRS: ADV.G.3

Difficulty Level = 2; Teaching Time = 5–8 minutes; Purpose = Example Ta nwdo 6o0f °t, hseo ttrhi ae nrgelme ’ as ianni nggl easnagrlee 3m0u° s t be 90°. Since this is a 30° − 60° − 90° ti sr ieaqnug al el , ttohtehsei dl eenogpt hp oosf i tt he et hhey p6 o0t°eannugslee multiplied by 2 3 : BC AC 2 3 3 & = = ^ h 6 & & = AC AC AC 3 3 6 h 2 3 . Alternatively, “eliminate-and-guess” based on the information provided in the �igure. Since AC is longer than BC (though nb eo ts bl i yg hmt luy cmh )o, rt eh et hcaonr r3e. c(tAa)nasnwde (r Bm) uc as tn ibme meleimdiiantaetleydbseinelcieminated. (D) can also 3 3 0 ] g is more than 1.5 times 3 ( 3 1.7 . ), and the length of AC is obviously less than 1.5 times the length of BC . Therefore, (C) must be the correct answer. 10. (H) Mathematics/Geometry/Triangles/ Properties of Triangles CC: 8.G.B.7 CCRS: ADV.G.2 2 3 3 6 3 ^ ^ h Difficulty Level = 2; Teaching Time = 5–8 minutes; Purpose = Example T�iors�tin�idndthtehpeelerinmgtehteorfof the triangle, PQ using the Pythagorean theorem: 8 & PR PQ QR 2 2 2 2 ^ ^ ^ h h h 4 64 16 48 & & PQ PQ 2 2 2 ^ ^ h h 4 = = = 3 PQ 48 16 3 ] g . Then, �ind the sum of the three sides: 4 8 4 12 4 3 3 . Ab el t es kr ni papt ievde liyf , tmh ea npyr oopf et rhtei easboofv3e 0s°t e p s c a n − 60° − 90° triangles are applied. Since PQR T is a roifgthhtet rhiyapnog tl ee nwuist eh lae ns igdteh ,eiqt umaul tsot boen ea- h a l f 30° − 60° − 90° triangle and QR is opposite the 30° angle. Therefore, 4 = PQ 3 and the perimeter is 12 4 3 + .

Don’t Forget This! In a 30° − 60° − 90° t o r p ia p n o g s l i e te , t t h h e e l 3 e 0 g ° ahnypgloeteenquusaelslethnegth multiplied by 2 1 ; t6h0e° laeng gol ep pe qo suiat el s tthhee hmyuplot itpelni euds eb yl e n g t h 2 3 .

A. 3 0 B. 2 C. 2 3 D.

3 3 0

= = =

10. In the �igure below, the perimeter of PQR T = ?

Properties of Triangles Items #10–11

F. 12 + 3 G. 12 2 + 3 H. 12 4 + 3 J. 28