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Triangle Classification and Distance Formula Practice

Practice problems on classifying triangles and using the distance formula to find midpoint and perpendicular bisectors.

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Triangle Classification and Distance Formula Practice

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  1. ( x2 – x1)2 +( y2 – y1)2 = (4– (–3))2 +(–1 – 5)2 = 85 d 49 +36 = = ANSWER The correct answer is C. EXAMPLE 1 Standardized Test Practice SOLUTION Let( x1, y1 ) = ( –3, 5)and( x2, y2 ) = ( 4, –1 ).

  2. Classify ∆ABCas scalene, isosceles, or equilateral. = 3 2 AB = (7 – 4)2 + (3 – 6)2 = 18 = 29 = 29 BC = (2 – 7)2 + (1 – 3)2 ANSWER AC = (2 – 4)2 + (1 – 6)2 BecauseBC = AC, ∆ABCis isosceles. EXAMPLE 2 Classify a triangle using the distance formula

  3. Find the midpoint of the line segment joining (–5, 1) and (–1, 6). ( ) x1 + x2y1 + y2 , 2 2 72 = (– 3, ) , ( ) – 5 + (–1) 1 + 6 = 2 2 EXAMPLE 3 Find the midpoint of a line segment SOLUTION Let( x1, y1 ) = (–5, 1)and( x2, y2 ) = (–1, 6 ).

  4. Write an equation for the perpendicular bisector of the line segment joining A(– 3, 4) and B(5, 6). STEP1 ( ) x1 + x2y1 + y2 , Find the midpoint of the line segment. 2 2 , ( ) – 3 + 5 4 + 6 = 2 2 EXAMPLE 4 Find a perpendicular bisector SOLUTION = (1, 5)

  5. Calculate the slope ofAB 6 – 4 = m = y2 – y1 = 1m 14 28 = 5 – (– 3) x2 – x1 STEP3 Find the slope of the perpendicular bisector: 1 1/4 – – = EXAMPLE 4 Find a perpendicular bisector STEP2 = – 4

  6. STEP4 Use point-slope form: An equation for the perpendicular bisector ofABisy = – 4x + 9. y = – 4x + 9. or ANSWER EXAMPLE 4 Find a perpendicular bisector y –5= – 4(x –1),

  7. Many scientists believe that an asteroid slammed into Earth about 65 million years ago on what is now Mexico’s Yucatan peninsula, creating an enormous crater that is now deeply buried by sediment. Use the labeled points on the outline of the circular crater to estimate its diameter. (Each unit in the coordinate plane represents 1 mile.) EXAMPLE 5 Solve a multi-step problem Asteroid Crater

  8. STEP 1 Write equations for the perpendicular bisectors of AOand OBusing the method of Example 4. Perpendicular bisector of AO Perpendicular bisector of OB EXAMPLE 5 Solve a multi-step problem SOLUTION y = – x + 34 y = 3x + 110

  9. STEP 2 Find the coordinates of the center of the circle, where AOand OBintersect, by solving the system formed by the two equations in Step 1. EXAMPLE 5 Solve a multi-step problem y= – x + 34 Write first equation. 3x + 110 = – x + 34 Substitute for y. 4x = – 76 Simplify. x = – 19 Solve for x. y = – (– 19) + 34 Substitute the x-value into the first equation. y = 53 Solve for y. The center of the circle isC (– 19, 53).

  10. STEP 3 Calculate the radius of the circle using the distance formula. The radius is the distance between Cand any of the three given points. OC = (–19 – 0)2 + (53 – 0)2 = 3170 56.3 ANSWER The crater has a diameter of about2(56.3) = 112.6miles. EXAMPLE 5 Solve a multi-step problem Use(x1, y1) = (0, 0)and(x2, y2) = (–19, 53).

  11. 18 18 STEP 1 Rewrite the equation in standard form. x = – EXAMPLE 1 Graph an equation of a parabola Graphx= – y2. Identify the focus, directrix, and axis of symmetry. SOLUTION Write original equation. Multiply each side by–8. – 8x = y2

  12. STEP 2 STEP 3 Identify the focus, directrix, and axis of symmetry. The equation has the form y2 = 4pxwhere p = – 2. The focus is (p, 0), or (– 2, 0). The directrix is x = – p, or x = 2. Because yis squared, the axis of symmetry is the x - axis. Draw the parabola by making a table of values and plotting points. Because p < 0, the parabola opens to the left. So, use only negative x - values. EXAMPLE 1 Graph an equation of a parabola

  13. EXAMPLE 1 Graph an equation of a parabola

  14. Write an equation of the parabola shown. 3 2 3 2 The graph shows that the vertex is (0, 0) and the directrix is y = – p = for pin the standard form of the equation of a parabola. – ( )y 32 x2= 4 Substitute for p EXAMPLE 2 Write an equation of a parabola SOLUTION x2 = 4py Standard form, vertical axis of symmetry x2 = 6y Simplify.

  15. The EuroDish, developed to provide electricity in remote areas, uses a parabolic reflector to concentrate sunlight onto a high-efficiency engine located at the reflector’s focus. The sunlight heats helium to 650°C to power the engine. EXAMPLE 3 Solve a multi-step problem Solar Energy •Write an equation for the EuroDish’s cross section with its vertex at (0, 0). •How deep is the dish?

  16. EXAMPLE 3 Solve a multi-step problem SOLUTION STEP 1 Write an equation for the cross section. The engine is at the focus, which is | p | = 4.5 meters from the vertex. Because the focus is above the vertex, pis positive, so p = 4.5. An equation for the cross section of the EuroDish with its vertex at the origin is as follows: x2 = 4py Standard form, vertical axis of symmetry x2 = 4(4.5)y Substitute 4.5 for p. Simplify. x2 = 18y

  17. Find the depth of the EuroDish. The depth is the y- value at the dish’s outside edge. The dish extends = 4.25 meters to either side of the vertex (0, 0), so substitute 4.25forxin the equation from Step 1. 8.5 2 vertex (0, 0), so substitute 4.25 for xin the equation from Step 1. 1.0 y EXAMPLE 3 Solve a multi-step problem STEP 2 x2 = 18y Equation for the cross section (4.25)2 = 18y Substitute4.25forp. Solve for y.

  18. ANSWER The dish is about 1 meter deep. EXAMPLE 3 Solve a multi-step problem

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