12 multiple-choice questions, progressively harder.
Let A(−8,13)A(-8,13)A(−8,13) and B(13,−1)B(13,-1)B(13,−1). Find internal P with AP:PB=3:4AP:PB=3:4AP:PB=3:4.
Solution
Correct answer: C
The full changes are 21 and -14. Three sevenths of them are 9 and -6.
A is (−4,6)(-4,6)(−4,6) and internal P is (2,−3)(2,-3)(2,−3). If AP:PB=2:3AP:PB=2:3AP:PB=2:3, find B.
Correct answer: A
The known changes 6 and -9 are two fifths of the whole changes. Multiply them by five halves before adding to A.
A is (15,−9)(15,-9)(15,−9), B is (−5,6)(-5,6)(−5,6), and P is (7,−3)(7,-3)(7,−3). What is the ordered internal ratio AP:PBAP:PBAP:PB?
Correct answer: D
The horizontal fraction is −8−20\frac{-8}{-20}−20−8 and the vertical fraction is 615\frac6{15}156. Both are two fifths and lie between zero and one.
A is (−1,8)(-1,8)(−1,8) and B is (9,−7)(9,-7)(9,−7). Which candidate is the internal point with AP:PB=3:2AP:PB=3:2AP:PB=3:2?
Correct answer: B
Use three fifths of the horizontal change 10 and vertical change -15.
The internal point P on A(2,11)A(2,11)A(2,11) to B(17,−4)B(17,-4)B(17,−4) has second coordinate 1. What is its first coordinate?
The vertical change to P is -10 out of the full -15, so the fraction is two thirds. The horizontal change must use the same fraction.
A is (6,−12)(6,-12)(6,−12) and B is (6,18)(6,18)(6,18). P has first coordinate 6 and AP:PB=4:1AP:PB=4:1AP:PB=4:1. What is yPy_PyP?
The changing coordinate moves four fifths of the way from -12 to 18.
For A(−5,−2)A(-5,-2)A(−5,−2) and B(10,13)B(10,13)B(10,13), P satisfies AP:PB=1:2AP:PB=1:2AP:PB=1:2. Starting from B, which fraction toward A reaches the same P?
From B the first part is BP, which has two of the three ratio parts. This takes B back to P=(0,3).
On segment AB, AP=18AP=18AP=18 cm and PB=30PB=30PB=30 cm. Which weights on A and B respectively give P?
The ordered ratio 18:30 reduces to 3:5. The opposite endpoint weights are five and three.
A straight beam has endpoints A(−9,2)A(-9,2)A(−9,2) and B(19,16)B(19,16)B(19,16) in meters. An internal support P must satisfy AP:PB=3:4AP:PB=3:4AP:PB=3:4. Where is P?
The support is three sevenths along the straight beam. Apply that fraction to the changes 28 and 14.
A is (5,4)(5,4)(5,4), internal P is (1,10)(1,10)(1,10), and AP:PB=1:3AP:PB=1:3AP:PB=1:3. Which coordinate equation correctly recovers yBy_ByB?
The ratio gives a weight of three on A and one on B, with denominator four.
Multiplying by four gives the required equation.
A is (−10,−3)(-10,-3)(−10,−3) and B is (5,12)(5,12)(5,12). P is (0,6)(0,6)(0,6). Is P on segment AB?
The horizontal fraction is 10/15, but the vertical fraction is 9/15.
One straight journey cannot use two different fractions. Being inside both coordinate ranges is insufficient.
An internal point is defined by AP:PB=8:12AP:PB=8:12AP:PB=8:12. Which ratio gives exactly the same point on the same ordered segment AB?
Divide both parts by the same positive number, four. Both ratios give a journey fraction of two fifths.
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