sat suite question viewer

Geometry and Trigonometry Difficulty: Medium

The figure presents triangle A, B C, where side A, C is horizontal and point B is above side A, C. Point D lies on side A, C, and line segment B D is drawn, forming right angle B D C. Side B C is labeled 12. Angle A, B D is labeled 30 degrees and angle B C D is labeled 60 degrees.

In $triangle A, B C$ above, what is the length of $segment A, D$ ?

Back question 108 of 268 Next

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268

Explanation

Choice B is correct. Triangles ADB and CDB are both $30 degree 60 degree 90 degree$ triangles and share $side B D$ . Therefore, triangles ADB and CDB are congruent by the angle-side-angle postulate. Using the properties of $30 degree 60 degree 90 degree$ triangles, the length of $side A, D$ is half the length of hypotenuse $A, B$ . Since the triangles are congruent, $the length of side A, B equals the length of side B C, which equals 12$ . So the length of $side A, D$ is $twelve halves equals 6$ .

Alternate approach: Since angle CBD has a measure of $30 degrees$ , angle ABC must have a measure of $60 degrees$ . It follows that triangle ABC is equilateral, so side AC also has length 12. It also follows that the altitude BD is also a median, and therefore the length of AD is half of the length of AC, which is 6.

Choice A is incorrect. If the length of $side A, D$ were 4, then the length of $side A, B$ would be 8. However, this is incorrect because $side A, B$ is congruent to $side B C$ , which has a length of 12. Choices C and D are also incorrect. Following the same procedures as used to test choice A gives $side A, B$ a length of $12 times the square root of 2$ for choice C and $12 times the square root of 3$ for choice D. However, these results cannot be true because $side A, B$ is congruent to $side B C$ , which has a length of 12.