EXAMPLE 1 Standardized Test Practice SOLUTION Let ( x 1, y 1 ) = ( –3, 5) and ( x 2, y 2 ) = ( 4, – 1 ). = (4 – (–3)) 2 + (– 1 – 5) 2 = = 85 ( x 2 – x 1 ) 2 + ( y 2 – y 1 ) 2 = d ANSWER The correct answer is C.

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

– 5 + (– 1) ( ) = 2 2, x 1 + x 2 y 1 + y 2 2 2, Find the midpoint of a line segment EXAMPLE 3 Let ( x 1, y 1 ) = (–5, 1) and ( x 2, y 2 ) = (– 1, 6 ). = ( – 3, ) 7272 Find the midpoint of the line segment joining (–5, 1) and (–1, 6). SOLUTION

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

Find a perpendicular bisector EXAMPLE 4 STEP 2 m = y 2 – y 1 x 2 – x 1 = 6 – 4 5 – (– 3) = 2828 = 1414 STEP 3 Find the slope of the perpendicular bisector: – 1m1m – = 1 1/4 = – 4 Calculate the slope of AB

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

Solve a multi-step problem EXAMPLE 5 Asteroid Crater 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.)

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

Solve a multi-step problem EXAMPLE 5 STEP 2 Find the coordinates of the center of the circle, where AO and OB intersect, by solving the system formed by the two equations in Step 1. y = – x + 34 Write first equation. 3x = – 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 is C (– 19, 53).

Solve a multi-step problem EXAMPLE 5 STEP 3 Calculate the radius of the circle using the distance formula. The radius is the distance between C and any of the three given points. OC = (–19 – 0) 2 + (53 – 0) 2 = Use (x 1, y 1 ) = (0, 0) and (x 2, y 2 ) = (–19, 53). ANSWER The crater has a diameter of about 2(56.3) = miles.

EXAMPLE 1 Graph an equation of a parabola SOLUTION STEP 1 Rewrite the equation in standard form x = – Write original equation Graph x = – y 2. Identify the focus, directrix, and axis of symmetry. – 8x = y 2 Multiply each side by – 8.

EXAMPLE 1 Graph an equation of a parabola STEP 2 Identify the focus, directrix, and axis of symmetry. The equation has the form y 2 = 4px where p = – 2. The focus is (p, 0), or (– 2, 0). The directrix is x = – p, or x = 2. Because y is squared, the axis of symmetry is the x - axis. STEP 3 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

EXAMPLE 2 Write an equation of a parabola SOLUTION The graph shows that the vertex is (0, 0) and the directrix is y = – p = for p in the standard form of the equation of a parabola – x 2 = 4py Standard form, vertical axis of symmetry x 2 = 4 ( ) y 3232 Substitute for p 3232 x 2 = 6y Simplify. Write an equation of the parabola shown.

EXAMPLE 3 Solve a multi-step problem Solar Energy 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. Write an equation for the EuroDish’s cross section with its vertex at (0, 0). How deep is the dish?

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, p is positive, so p = 4.5. An equation for the cross section of the EuroDish with its vertex at the origin is as follows: x 2 = 4py Standard form, vertical axis of symmetry x 2 = 4(4.5)y Substitute 4.5 for p. x 2 = 18y Simplify.

EXAMPLE 3 Solve a multi-step problem STEP 2 x 2 = 18y Equation for the cross section (4.25) 2 = 18y Substitute 4.25 for p. Solve for y. 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 4.25 for x in the equation from Step vertex (0, 0), so substitute 4.25 for x in the equation from Step y

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