Tip #26 to Get a Top ACT Math Score (page 2)
I shall call him … "Mini-Me."
Dr. Evil, Austin Powers: The Spy Who Shagged Me (New Line Cinema, 1999)
Similar triangles are two triangles where one is a shrinky version of the other, like Dr. Evil and Mini-Me. Since one is a shrunken version of the other, all sides are proportional, shortest to shortest and longest to longest. The ACT loves similar triangles.
Let's have a look at this question:
Solution: Here's your chance to see if you really learned Skill 25. We have to determine the measures of sides of triangle PQR shown above. Fill in the sides of special right triangle ΔPQR: 3, , 6. Once you have the lengths of the sides of ΔPQR, we can determine the lengths of ΔMNO. Triangle MNO is "similar" to triangle PQR, which means that the sides are proportional. So the ratio of the smallest sides must equal the ratio of the longest sides. Fill in the numbers and cross-multiply to solve. (Remember Mantra #21, "When you see a proportion, cross-multiply.")
So 3x = 27 and x = 9.
Correct answer: H
- In the figure below, ΔXYZ and ΔPQR are similar triangles with the side lengths given in feet. What is the perimeter, in feet, of ΔXYZ ?
- Isosceles triangle ΔABC is similar to triangle ΔMNO. If AC = 4, AB = 5, BC = 4, and MN = 15, what is the measure of MO ?
- A person 2 meters tall casts a shadow 4 meters long. At the same time a lamppost casts a shadow 15 meters long. How many meters tall is the lamppost?
- Triangle ΔPQR is similar to triangle ΔXYZ. is 6 centimeters, is 8 centimeters, and is 14 centimeters. If the longest side of ΔXYZ is 35 centimeters, what is the perimeter, in centimeters, of ΔXYZ ?
- In the figure below, A is on and B is on . What is the length of ?
- E To get the perimeter of ΔXYZ, we just add up the lengths of the sides. We are missing one side, which we will get by using similar triangles. How did I know? Because the question told me. If it mentions similar triangles, you will use them, guaranteed. That's how the ACT works. So when it says similar triangles, set up a proportion and solve for a side. The letters of the triangles will be written in corresponding order, so XY corresponds to PQ, since they are the first two letters in the order that each triangle is written, ΔXYZ and ΔPQR. Since similar triangles have proportional sides, to solve for XY, set up the proportion:
- H When a picture is described, draw it. Triangle ABC is isosceles, with AC Δ BC. Similar triangles have proportional sides, and their letters are written in corresponding order. So since MN corresponds to AB, and MO corresponds to AC, we can set up the proportion:
- B When a picture is described, draw it. This shows you that we have two triangles, which are similar. Set up a proportion:
- G When a picture is described, draw it. We want the perimeter of ΔXYZ, and we only have the longest side, which must be XZ since it corresponds to the longest side of ΔPQR (we can tell by the order of the letters in ΔXYZ and ΔPQR). So we set up proportions to solve for each side of ΔXYZ:
- C If two triangles have two equal angles, then they are similar. Basically, on the ACT if two triangles seem similar, they probably are. So since they are similar, their sides are proportional, smallest to smallest and largest to largest. Let's say BY = x, then ZY = 450 + x. Set up the proportions and cross-multiply:
Cross-multiply to get 144 = 6XY. Divide both sides by 6 to get 24 = XY. Thus, the perimeter of ΔXYZ = 24 + 18 + 27 = 69.
Cross-multiply to get 60 = 5(MO). Divide by 5 to get MO = 12.
When you see a proportion, cross-multiply: 30 = 4x. And divide by 4 to get 7.5.
Yoga and The Society for Free Living
Most kids had trouble with fractions back in 3rd grade. This makes sense, since an 8-year-old's brain is not usually ready to deal with fractions. Nine years later, however, even though a 17-year-old's brain is more than ready, many teenagers still believe that they "suck" at fractions. They are unknowingly holding on to an outdated and unnecessary belief.
Yoga can help you get past pesky outdated fears. It opens blocked pathways in the brain and body, dropping the clouded lens of prior beliefs. This allows you to see clearly and experience reality as it truly is.
Here is a yoga sequence designed for you by Corinne Andrews, a yoga teacher in western Massachusetts. Corinne designed the sequence to help you relieve stress, get energized, and focus your mind. We all have the potential to release fear, trust ourselves, and live freely.
Record these instructions into your iPod. Then find a comfortable spot and get yoga-ing. If you can't say "buttocks" without giggling, visit www.brianleaf.com to download a free podcast of Corinne leading you through the poses.
- Alternate Nostril Breathing
- Skull Shining Breath
- Cat Lift and Round
- Half Sun Salute
- Downward Dog
- Fish Pose
- Relaxation Pose
Sit in a chair or on the floor with a blanket or pillow underneath your bum. Close off your right nostril with your thumb, and inhale slowly through the left nostril. Then close off your left nostril with your ring finger, and exhale slowly through the right. Then inhale through right, switch fingers, and exhale through left. Repeat this for a few minutes. This breath balances the right and left hemispheres of your brain. You can do it anytime, even during the test. If you get a strange look, just explain that you are balancing your hemispheres.
Exhale sharply out through your nose as if blowing out a candle, and then let the inhale come in naturally. Repeat for 10 breaths, and then take one relaxed deep breath. Repeat the process two times. This clears your mind—and your sinuses.
Begin on your hands and knees, so you look like a coffee table. Inhale as you look up, allowing your back to arch down, and then slowly exhale as you look down, rounding your back up. Repeat 10 times. This practice gently warms up the spine and nervous system, and relaxes the upper back and shoulders.
Stand with your feet together. Inhale and lift your arms up, then exhale, bending your knees a bit, and slowly fold over to touch the floor. Inhale as you slide your hands up your legs to come up halfway, and then exhale to fold back down. Finally, inhale to come all the way back up. This practice warms up your whole body and helps you connect to the rhythm of your breath.
Begin on all fours, with your hands shoulder-width apart. Press firmly down through all parts of your fingers and hands, and reach your tailbone (bum) into the air. Your body should resemble an upside-down V. Keep your knees a bit bent and reach your bum way up. This pose invigorates your entire body and mind, while also relaxing your shoulders and upper back. Stay in this posture for 3 relaxed breaths. Then slowly lower down.
Sit in a chair and gently arch your upper back over the back of the chair. Breathe 3 relaxed breaths. This pose opens and stretches your chest, shoulders, and neck. You can also do this pose by placing a rolled blanket or towel underneath your shoulder blades as you lay on the floor.
Lie on your back on a carpet or mat. Take a few deep breaths, allowing each exhalation to be a long sigh. Allow your body to relax and to be supported by the floor. Relax as your thoughts pass through your mind. Do not engage with them, just witness. (You can sample a few minutes of relaxing music here, probably not DMX.)
(After a few minutes … )
Begin to deepen the breath. … Feel your belly and chest rise and fall with each breath. … Wiggle your fingers and toes. Then gently roll to one side and come to a seated position. Notice how you feel. Set the intention to take this feeling into your day, into your relationships, into your schoolwork, into your ACT prep. You've entered The Society For Free Living.
Go to: Tip #27
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